Geometric analysis and applications, Proc. nat. res. symposium, Canberra

Citation preview

PREFACE This volume contains the proceeding of the National Research Symposium on Geometric Analysis and Applications held at the Centre for Mathematics and its Applications, Australian National University, Canberra, from June 26 – 30, 2000. The Symposium celebrated the many significant contributions of Professor Derek W. Robinson to mathematics, on the occasion of his 65th birthday. The first day, Monday June 26, in particular was devoted to Derek; speakers with a particularly close connection to Derek, including A. Carey, M. Cowling, D. Evans, P. Jorgensen and T. ter Elst recalled and elaborated on important aspects of Derek’s work, and the day ended with a banquet in Derek’s honour. The Symposium brought together researchers working in harmonic analysis, linear and nonlinear partial differential equations, quantum mechanics and mathematical physics, and included researchers from North America, Europe and Asia as well as Australasia. We gratefully acknowledge the support of the contributors to this volume. We would like especially thank to Derek W. Robinson for his participation. A short synopsis of Derek’s career, and a full list of his publications to date are included in this volume. Contributions for the proceedings were sought from all participants and all papers received were carefully refereed by peer referees.

Alexander Isaev, Andrew Hassell, Alan McIntosh, Adam Sikora (Editors)

i

At the end of 2000 Derek Robinson retired from his position as Professor of Mathematics in the Institute of Advanced Studies at ANU, a position he had held for 19 years. Robinson obtained his early training at Oxford with a Bachelor’s degree in mathematics in 1957 and a Doctorate of Philosophy in theoretical nuclear physics in 1960. Subsequently he held various postdoctoral positions in Switzerland, the United States, Germany and France before being appointed Professor of Mathematical Physics at the University of Aix–Marseille in 1968. He remained in this position until 1978 at which point he accepted appointment as Professor of Mathematics at the University of New South Wales. Soon after he moved to Australia he became a Fellow of the Australian Academy of Science. In his early years Robinson worked in a wide range of areas of mathematical physics, quantum field theory, statistical mechanics, operator algebras, etc. Operator algebras were introduced into quantum field theory in the 1960s as a means to describe the macroscopic observables but their most fruitful application was to the characterization of the equilibrium states of statistical mechanics. The observables of classical statistical mechanics form an abelian C ∗ -algebra and Robinson realized that the corresponding quantum algebra was ‘asymptotically abelian’. This observation was of fundamental importance since the states of asymptotically abelian algebras could be shown to share many of the good properties of abelian algebras. Therefore extremal states of the algebra could be identified with pure phases of the system and decomposition theory of the states could be applied to describe the separation of the phases. Although these results were of a general abstract nature Robinson showed that they could be applied to a broad class of realistic models of quantum spin systems. In particular he demonstrated that a number of these systems exhibited a phase transition at low temperatures and the pure phases were indeed described by extremal invariant states. Much of this is described in the two volume monograph Operator Algebras and Quantum Statistical Mechanics which Derek Robinson co-authored with Ola Bratteli. Both volumes of this book appeared in a second edition and continue to be used by research workers in these areas twenty years later. The majority of applications of operator algebras to quantum statistical mechanics concerned the equilibrium theory. Robinson and Bratteli realized that the description of non-equilibrium phenomena required a dynamical theory and this in turn required a theory of unbounded derivations of the algebras. The foundations of this theory were laid in a series of joint papers which are also described in their book. As a direct result of this latter work Robinson’s interests then developed to evolution equations and semigroup ii

theory in a much broader context. This change of direction coincided with his move to Australia and most of his subsequent work has been on evolution equations, in particular equations involving elliptic operators on Lie groups. One of the issues at that time was the integrability of a Lie algebra representation. Together with Ola Bratteli, George Elliott and Palle Jørgensen, Robinson gave a sufficient condition in terms of a dissipativity condition and an estimate on the semigroup generated by a Laplacian. Using Lipschitz spaces Robinson could not only weaken the assumptions in the above mentioned paper, but he could also prove that the analytic vectors associated with any representation of a Lie group coincide with the analytic elements of the Poisson semigroup. By this stage Robinson became more interested in elliptic operators on Lie groups and decided to write the monograph Elliptic Operators and Lie Groups. After completion of the book Robinson began a collaboration with Tom ter Elst and their continuing work can be divided into three parts. First, they developed the theory of complex, weighted, higher order subcoercive operators on Lie groups, in particular proving Gaussian bounds for the kernel, and its derivatives, of the semigroup generated by such an operator. They established that the Gaussian bounds are equivalent with a subcoercivity condition on the operator, under weak additional assumptions. Secondly, they studied second-order divergence form subelliptic operators on Lie groups with complex bounded measurable coefficients and proved, under a variety of conditions, optimal smoothness properties of the kernel. Thirdly, in joint work with Adam Sikora or Nick Dungey, they studied asymptotic properties of the semigroup and its kernel. They showed that on a Lie group with polynomial growth the second-order Riesz transforms associated with the Laplacian are bounded if, and only if, the group is a direct product of a compact and a nilpotent Lie group. Their analysis of the asymptotics of higher order operators continues. Robinson has maintained an active involvement in University affairs, having been Chairman of the Board of the Institute for Advanced Studies at ANU (1988-1992), and a member of ANU Council (1997-2000), where he never resiled from robust and provocative debate. He has also been an active member of the mathematical community, in particular being president of the Australian Mathematical Society (1994-1996) and continuing as Vicepresident. He also served four years as a member of the Research Training and Careers Committee of the ARC (1996-2000) and as Chair of the National Committee for Mathematics (1997-2001).

iii

LIST OF PUBLICATIONS Books [1] The Thermodynamic Pressure in Quantum Statistical Mechanics - Springer Verlag Lecture Notes in Physics 9, (1971) 115 pages. [2] Operator Algebras and Quantum Statistical Mechanics Vol I (with O Bratteli) 500 pages, Springer-Verlag (1979), 2nd edition 505 pages, Springer-Verlag (1979). Russian translation of first edition (1981). [3] Operator Algebras and Quantum Statistical Mechanics Vol II (with O Bratteli) 501 pages, Springer-Verlag (1981), 2nd edition 517 pages, Springer-Verlag (1996). [4] Basic Semigroup Theory Proceedings of the Centre for Mathematical Analysis, ANU, Canberra Volume 2, (1983) 138 pages. [5] Elliptic Operators and Lie Groups 556 pages, Oxford University Press (1991).

Research papers [1] Multiple Coulomb Excitations of Vibrational Nuclei Nuclear Physics 45 (1961) 459–470. [2] Zero-Mass Representations of the Inhomogeneous Lorentz Group Helv Phys Acta 35 (1962) 98–112. [3] Support of a Field in Momentum Space Helv Phys Acta 35 (1962) 403–413. [4] Multiple Coulomb Excitations of Deformed Nuclei Helv Phys Acta 36 (1963) 140–154. [5] On a Soluble Model of Relativistic Field Theory Physics Letters 9 (1964) 189–191. [6] A Theorem Concerning the Positive Metric Commun Math Phys 1 (1965) 89–95. [7] The Ground State of the Bose Gas Commun Math Phys 1 (1965) 159–174. [8] Conserved Currents and Associated Symmetries (with D Kastler and A Swieca) Commun Math Phys 2 (1966) 108–120. [9] Covariance Algebras in Field Theory (with S Doplicher and D Kastler) Commun Math Phys 3 (1966) 1–28. [10] Invariant States in Statistical Mechanics (with D Kastler) Commun Math Phys 3 (1966) 151–180. [11] Extremal Invariant States (with D Ruelle) Ann Inst Henri Poincar´ e 6 (1967) 299-310. [12] Mean Entropy of States of Classical Statistical Mechanics (with D Ruelle) Commun Math Phys 5 (1967) 288–300. [13] Mean Entropy of States of Quantum Statistical Mechanics (with O E Lanford) J Math Phys 9 (1968) 1120–1125. [14] Statistical Mechanics of Quantum Spin Systems Commun Math Phys 6 (1967) 151–160. [15] Asymptotic Abelian Systems (with S Doplicher, R Kadison and D Kastler) Commun Math Phys 6 (1967) 101–120. [16] Analyticity Properties of a Lattice Gas (with G Gallavotti and S Miracle-Sole) Physics Letter 25A (1967) 443–444. [17] Statistical Mechanics of Quantum Spin Systems II Commun Math Phys 7 (1968) 337–346. [18] Statistical Mechanics of Quantum Spin Systems III (with O E Lanford) Commun Math Phys 9 (1968) 327–338. [19] Analyticity Properties of the Anisotropic Heisenberg Model (with G Gallavotti and S Miracle-Sole) Commun Math Phys 10 (1968) 311–324. [20] Proof of the Existence of Phase Transitions in the Anisotropic Heisenberg Model Commun Math Phys 14 (1969) 196–204. [21] Physical States of Fermi Systems (with S Miracle-Sole) Commun Math Phys 14 (1969) 235–270. [22] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores I (with S MiracleSole) Commun Math Phys 16 (1970) 290–309. [23] Statistical Mechanics of Quantum Mechanical Particles with Hard Cores II (with S MiracleSole) Commun Math Phys 19 (1970) 204–218. [24] Normal and Locally Normal States Commun Math Phys 19 (1970) 219–234. [25] Normal States and Representations of the Canonical Commutation Relations (with M Courbage and S Miracle-Sole) Ann Inst Henri Poincare XIV (1971) 171–178. iv

[26] Approach to Equilibrium of Free Quantum Systems (with O E Lanford) Commun Math Phys 24 (1972) 193–210. [27] The Finite Group Velocity of Quantum Spin Systems (with E H Lieb) Commun Math Phys 28 (1972) 193–210. [28] Return to Equilibrium Commun Math Phys 31 (1973) 171-189. [29] Scattering Theory with Singular Potentials I Ann Inst Henri Poincare XXI (1974) 185–215. [30] Scattering Theory with Singular Potentials II (with P Ferrero and O Depazzis) Ann Inst Henri Poincare XXI (1974) 216–231. [31] A Characterisation of Clustering States Commun Math Phys 41 (1975) 79–88. [32] Dynamical Stability and Pure Thermodynamic Phases (with H Narnhofer) Commun Math Phys 41 (1975) 89–97. [33] Unbounded Derivations of C ∗ -Algebras I (with O Bratteli) Commun Math Phys 42 (1975) 253–268. [34] Unbounded Derivations of C ∗ -Algebras II (with O Bratteli) Commun Math Phys 46 (1976) 11–30. [35] Unbounded Derivations of Von Neumann Algebras (with O Bratteli) Ann Inst Henri Poincare XXV (1976) 139–164. [36] Unbounded Derivations and Invariant Trace States (with O Bratteli) Commun Math Phys 46 (1976) 31–35. [37] The Approximation of Flows J Funct Anal 24 (1977) 280–290. [38] Green’s Functions, Hamiltonians and Modular Automorphisms (with O Bratteli) Commun Math Phys 50 (1976) 53–59. [39] Bose-Einstein Condensation with Attractive Boundary Conditions Commun Math Phys 50 (1976) 53–59. [40] Properties of Propagation of Quantum Spin Systems J Aust Math Soc Vol XIX (Series B) 19 (1976) 387–399. [41] Quasi Analytic Vectors and Derivations of Operator Algebras (with O Bratteli and R Herman) Math Scan 39 (1976) 371–38. [42] Perturbations of Flows on Banach Spaces and Operator Algebras (with O Bratteli and R Herman) Commun Math Phys 59 (1978) 167–196. [43] Stability Properties and the KMS Condition (with A Kishimoto and O Bratteli) Commun Math Phys 61 (1978) 209–238. [44] Ground States of Quantum Spin Systems (with A Kishimoto and O Bratteli) Commun Math Phys 64 (1978) 41–48. [45] Propagation Properties in Scattering Theory J Aus Math Soc (Series B) 21 (1979) 474–485. [46] Lie and Jordan Structure in Operator Algebras (with E Stormer) J Aus Math Soc (Series A) 29 (1980) 129–142. [47] Equilibrium States of a Bose Gas with repulsive interactions (with O Bratteli) J Aus Math Soc (Series B) 22 (1980) 129–147. [48] Positivity and Monotonicity Properties of C 0 -Semigroups I (with O Bratteli and A Kishimoto) Commun Math Phys 75 (1980) 67–84. [49] Positivity and Monotonicity Properties of C 0 -Semigroups II (with A Kishimoto) Commun Math Phys 75 (1980) 85–101. [50] Subordinate Semigroups and Order Properties (with A Kishimoto) J Aus Math Soc (Series A) 31 (1981) 69–76. [51] Positive C0 -Semigroups on C ∗ -Algebras (with O Bratteli) Math Scand 49 (1981) 259–274. [52] Strictly Positive and Strongly Positive Semigroups (with A Majewski) J Aus Math Soc (Series A) 34 (1983) 36–48. [53] Strongly Positive Semigroups and Faithful Invariant States Commun Math Phys 85 (1982) 129–142. [54] On Unbounded Derivations Commuting with a Compact Group of ∗-Automorphisms (with A Kishimoto) Publ RIMS Kyoto Univ 18 (1982) 1121–1136. [55] Positive Semigroups on Ordered Banach Spaces (with O Bratteli and T Digernes) J Op Theor 9 (1983) 371–400. [56] Addition of an Identity to an Ordered Banach Space (with S Yamamuro) J Aus Math Soc (Series A) 35 (1983) 200–210. [57] Hereditary Cones, Order Ideals, and Half-norms (with S Yamamuro) Pac J Math 110 (1984) 335–343. v

[58] The Jordan Decomposition and Half-norms (with S Yamamuro) Pac J Math 110 (1984) 345–353. [59] The Canonical Half-norm, Dual Half-norms, and Monotonic Norms (with S Yamamuro) Tohoku J Math 35 (1983) 375–386. [60] Continuous Semigroups on Ordered Banach Spaces J Funct Anal 5 (1983) 268–284. [61] On Positive Semigroups Publ RIMS Kyoto Univ 20 (1984) 213–224. [62] Extending Derivations (with C J K Batty, A Carey and D E Evans) Publ RIMS Kyoto Univ 20 (1984) 119–139. [63] Positive One-parameter Semigroups on Ordered Banach Spaces (with C J K Batty) Acta Appl Math 2 (1984) 221–296. [64] A C ∗ -algebraic Schoenberg Theorem (with O Bratteli, P Jørgensen and A Kishimoto) Ann Inst Fourier (Grenoble) XXXIV (1984) 155–187. [65] Relative Locality of Derivations (with O Bratteli and T Digernes) J Funct Anal 59 (1984) 12–40. [66] Derivations, Dynamical Systems, and Spectral Restrictions (with A Kishimoto) Math Scand 56 (1985) 83–95. [67] Dissipations, Derivations, Dynamical Systems and Asymptotic Abelianness (with A Kishimoto) J Op Theor 13 (1985) 237–253. [68] Derivations of Simple C ∗ -algebras Tangential to Compact Automorphism Groups (with E Stormer and M Takesaki) J Op Theor 13 (1985) 189–200. [69] Strong Topological Transitivity and C ∗ -dynamical Systems (with O Bratteli and G A Elliott) J Math Soc Japan 37 (1985) 115–133. [70] The Characterisation of Differential Operators by Locality; Classical Flows (with O Bratteli and G A Elliott) Comp Math 58 (1985) 279–319. [71] The Characterisation of Differential Operators by Locality; Abstract Derivations (with C J K Batty) Erg Theor and Dyn Syst 5 (1985) 171–183. [72] The Characterisation of Differential Operators by Locality; Dissipations and Ellipticity (with O Bratteli and G A Elliott) Publ RIMS Kyoto Univ 21 (1985) 1031–1049. [73] The Characterization of Differential Operators by Locality: C ∗ -algebras of Type I (with O Bratteli and G A Elliott) J Op Theor 16 (1986) 213–233. [74] Integration in Abelian C ∗ -dynamical Systems (with O Bratteli, T Digernes and F M Goodman) Publ RIMS Kyoto Univ, 21 (1985) 1001–1030. [75] Smooth Cores of Lipschitz Flows Publ RIMS Kyoto Univ 22 (1986) 659–669. [76] Smooth Derivations on Abelian C ∗ -dynamical Systems J Aus Math Soc (Series A) 42 (1987) 247–264. [77] Commutators and Generators (with C J K Batty) Math Scand 62 (1988) 303–326. [78] Commutators and Generators II Math Scand 64 (1989) 87–108. [79] Return to Equilibrium in the X −Y Model (with L R Hume) J Stat Phys 44 (1986) 829–848. [80] Embedding Product Type Actions into C ∗ -dynamical Systems (with O Bratteli and A Kishimoto) J Funct Anal 75 (1987) 188–210. [81] Fractional Powers of Generators of Equicontinuous Semigroups and Fractional Derivatives (with O E Lanford) J Aus Math Soc (Series A) 46 (1989) 1–32. [82] Commutator Theory on Hilbert Space Can J Math 5 (1987) 1235–1280. [83] The Differential and Integral Structure of Continuous Representations of Lie Groups J Op Theor 19 (1988) 95–128. [84] The Heat Semigroup and Integrability of Lie Algebras (with O Bratteli, F M Goodman and P E T Jørgensen) J Funct Anal 79 (1988) 351–397. [85] An Index for Continuous Semigroups of ∗ -endomorphisms of B(H) (with R T Powers) J Funct Anal 83 (1989) 1–12. [86] Lie Groups and Lipschitz Spaces Duke Math Jour 57 (1988) 357–395. [87] The Heat Semigroup, Derivations, and Reynold’s Identity (with O Bratteli and C J K Batty) Operator algebras and applications. London Math Soc Lecture Notes Series 136 (1988) 22–47. [88] The Heat Semigroup and Integrability of Lie Algebras: Lipschitz Spaces and Smoothness Properties Commun Math Phys 132 (1990) 217–243. [89] Lipschitz Operators J Funct Anal 85 (1989) 179–211. [90] Comparison of Commuting One-parameter Groups of Isometries (with O Bratteli and H Kurose) Trans Amer Math Soc 320 (1990) 677–694. vi

[91] Unitary Representations of Lie Groups and G˚ arding’s Inequality (with O Bratteli, F M Goodman and P E T Jørgensen) Proc Amer Math Soc 107 (1989) 627–632. [92] Positive Semigroups Generated by Elliptic Operators on Lie Groups (with W Arendt and C J K Batty). J Op Theor 23 (1990) 369–407. [93] Elliptic Differential Operators on Lie Groups J Funct Anal 97 (1991) 373–402. [94] 2nd order Elliptic Operators and Heat Kernels on Lie Groups (with O Bratteli) Trans Amer Math Soc 325 (1991) 683–713. [95] Subelliptic Operators on Lie Groups: Regularity (with A F M ter Elst) J Aus Math Soc 57 (1994) 179–229. [96] Subelliptic Operators on Lie Groups: Variable Coefficients (with O Bratteli) Acta Appl Math 42 (1996) 1–104. [97] Subcoercivity and Subelliptic Operators on Lie Groups I: Free Nilpotent Groups (with A F M ter Elst) Pot An 3 (1994) 283–387. [98] Subcoercivity and Subelliptic Operators on Lie Groups II:The General Case (with A F M ter Elst) Pot An 4 (1995) 205–243. [99] Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients (with A F M ter Elst) Publ RIMS Kyoto Univ 5 (1993) 745–801. [100] Lp -regularity of Subelliptic Operators on Lie Groups (with R J Burns and A F M ter Elst) J Op Theor 31 (1994) 165–187. [101] Functional Analysis of Subelliptic Operators on Lie Groups (with A F M ter Elst) J Op Theor 31 (1994) 277–301. [102] Asymptotics of Periodic Subelliptic Operators (with C J K Batty, O Bratteli and P E T Jørgensen) J Geom Anal 5 (1995) 427–443. [103] Weighted Strongly Elliptic Operators on Lie Groups (with A F M ter Elst) J Funct Anal 125 (1994) 548–603. [104] On Positive Rockland Operators (with P Auscher and A F M ter Elst) Colloq Math LXVII (1994) 197–216. [105] Spectral Estimates for Positive Rockland Operators (with A F M ter Elst) Algebraic groups and Lie groups (Ed G I Lehrer), Aus Math Soc Lecture Series 9. Cambridge Univ Press (1997). [106] Reduced Heat Kernels on Nilpotent Lie Groups (with A F M ter Elst) Commun Math Phys 173 (1995) 475–511. [107] Semigroup Kernels, Poisson bounds and Holomorphic Functional Calculus (with X T Duong) J Funct An 142 (1996) 89–129. [108] Weighted Subcoercive Operators on Lie Groups (with A F M ter Elst) J Funct Anal 156 (1998) 88–163. [109] On Kato’s Square Root Problem (with A F M ter Elst) Hokkaido Math J 26 (1997) 1–12. [110] Analytic Elements of Lie Groups (with A F M ter Elst) Helv Phys Acta 69 (1996) 655–678. [111] Abundance of Invariant and Almost Invariant Pure States of C ∗ -dynamical Systems (with O Bratteli and A Kishimoto) Commun Math Phys 187 (1997) 491–507. [112] Second-order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients (with A F M ter Elst) Acta Appl Math 59 (1999) 299–331. [113] High Order Divergence-form Elliptic Operators on Lie Groups (with A F M ter Elst) Bull Aus Math Soc 55 (1997) 335-348. [114] Second-order Subelliptic Operators on Lie Groups II: Real Measurable Coefficients (with A F M ter Elst) Progess in Nonlinear Differential Equations 42 Birkh¨ auser Verlag (2000) 103–124. [115] Second-order Strongly Elliptic Operators on Lie Groups with H¨ older Continuous Coefficients. (with A F M ter Elst) J Aus Math Soc (Series A) 63 (1997) 297–363. [116] Heat Kernels and Riesz Transforms on Nilpotent Lie Groups. (with A F M ter Elst and A Sikora) Colloq Math LXXIV (1997) 191–218. [117] Second-order Subelliptic Operators on Lie Groups III: H¨ older Continuous Coefficients (with A F M ter Elst) Calc Var P D E 8 (1999), 327–363. [118] Spectral Asymptotics of Periodic Elliptic Operators (with O Bratteli and P E T Jørgensen) Math Z 232 (1999), 621–650. [119] Local Lower Bounds on Heat Kernels (with A F M ter Elst) Positivity 2 (1998) 123–151. [120] Riesz Transforms and Lie Groups of Polynomial Growth. (with A F M ter Elst and A Sikora) J Funct Anal 162 (1999), 14–51. vii

[121] Asymptotics of Semigroups on Nilpotent Lie Groups (with N Dungey, A F M ter Elst and A Sikora) J Op Theor (to appear) [122] Asymptotics of Sums of Subcoercive Operators (with N Dungey and A F M ter Elst) Colloq Math 82 (1999) 231–260. [123] On second-order Periodic Elliptic Operators in Divergence Form (with A F M ter Elst and Adam Sikora) Math Z (to appear) [124] On Anomalous Asymptotics of Heat Kernels (with A F M ter Elst) Evolution equations and their applications to Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, Vol. 215, (2001) 89–103, Eds. G Lumer, L Weis, Marcel Dekker, New York [125] On Second-order Almost-periodic Elliptic Operators (with N Dungey and A F M ter Elst) J London Math Soc (submitted) [126] On Anomalous Asymptotics of Heat Kernels on Groups of Polynomial Growth (with N Dungey and A F M ter Elst) Comp Math (submitted)

Lecture notes, reviews and conference papers [1] Algebraic Aspects of Relativistic Field Theory - Brandeis Lectures, Gordon and Breach, New York, Vol 1 (1966). [2] Symmetries, Broken Symmetries, Charges and Currents - Istanbul Lecture, Freeman and Co, San Francisco, (1967) 463–513. [3] Conserved Currents and Broken Symmetries - Colloque du CNRS (1966), Eds du CNRS 15 (1968) 131–135. [4] Physical States and Finiteness Restrictions - Proceeding of the CNRS Conference at Gifsur-Yvette, CNRS (1969). [5] Existence Theorems in Quantum Statistical Mechanics - Proceedings of the Cargese Summer School (notes by R Lima and S Miracle) Gordon and Breach (1970). [6] Approach and Return to Equilibrium of Free Fermi States - Colloquium of the American Math Soc 1971 - Siam AMS Proceeding 5 55 (1972). [7] C ∗ -Algebras and Quantum Statistical Mechanics - Int School of Physics Enrico fermi Varenna-Editrice Comp Bologna (1975) 235–252. [8] Time Dependent Scattering Theory III - Int Colloq on Group Theoretical Methods, Marseille (1974). [9] Unbounded Derivations of C ∗ -Algebras - Int Symp on Math Problems in Theor Phys, Springer-Verlag (1975). [10] Dynamics in Quantum Statistical Mechanics - Int Summer School Bielefeld 1976, Plenum Press (1978). [11] Review of Derivations - Int Conference on Mathematical Physics, Lausanne 1979, SpringerVerlag (1980). [12] Commutator Theory and Partial Differential Operators on Hilbert Space - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 14 (1986) 295–302. [13] Differential Operators on C ∗ -algebras - Contemporary Mathematics, Amer Math Soc 62 (1987) 367–384. [14] C ∗ -algebras and a Single Operator - Surveys of some recent results in operator theory II Pitman Research 171 (1988) 235–266. [15] Integration of Lie Algebras - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 15 (1987) 255–278. [16] Schauder Estimates on Lie Groups- Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 24 (1989) 1–9. [17] Subelliptic Operators on Lie Groups (with A F M ter Elst) - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 29 (1992) 63–72. [18] Strongly Elliptic and Subelliptic Operators on Lie Groups - Quantum and NonCommutative Analysis, Eds H Araki et al., Kluwer Academic Publishers (1993) 435–453 [19] Basic Semigroup Theory - Proceedings of the Centre of Mathematical Analysis, ANU, Canberra 34–Part III (1996) 1–33. [20] Elliptic Operators on Lie Groups (with A F M ter Elst) Acta Appl. Math. 44 (1996) 133– 150. [21] C ∗ -Algebras and Mathematical Physics - Fields Institute Monograph volume, Eds R Bhat, G Elliott and P Fillmore, American Mathematical Society, Providence, Rhode Island. viii

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE HILARY BOOTH

Abstract. The Maxwell-Dirac equations give a model of an electron in an electromagnetic (e-m) eld, in which neither the Dirac or the e-m elds are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial dierential equations (PDE's). In this way the self- eld of the electron is included. We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its rst derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter. In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon eld must be equal and opposite to the central (external) eld. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die o exponentially with increasing distance from the central charge.

1. The Maxwell-Dirac Equations and QED The coupled Maxwell-Dirac equations can be written as follows:

 (@ , ieA ) + im = 0 where  = 0; : : : ; 3 F = A ; , A;  @ F = ,4ej where j =  : (1) Note that 2 C 4 is the Dirac wave-function or 4-spinor and the Dirac conjugate. These are acted upon by  which are the usual gamma matrices (representations of a Cliord algebra), and A is the 4-potential. These equations model an electron in an electromagnetic eld. The two equations are coupled via the Dirac current j i.e. we include 1

2

HILARY BOOTH

the nonlinearity of the self- eld, see Equation (1). These equations form the foundation of quantum electrodynamics (QED), the theory of electrons interacting with elds. QED is one of the most successful physical theories, explaining the Lamb shift and the anomalous magnetic moment of the electron. The calculations of QED are acheived by quantizing the elds and using perturbation theory. In so doing, wellknown mathematical problems occur, which have yet to be resolved at the fundamental level. It is possible that the full nonlinearized equations must be analysed rigorously before we can hope to resolve these deep problems. For example, in [23], Lieb and Loss observe that the stability of matter requires that the electron be de ned with a Dirac operator with the magnetic vector potential A(x), instead of the free Dirac operator (without A(x)). That perturbation theory must start from the dressed electrons (including their own self- eld) might be \fundamentally important in a non-perturbative QED". This view is shared by many analysts working on this problem including Flato, Simon and Ta in who established global existence for the M-D equations as recently as 1997 [18], following many years of sustained interest [21] [9] [20] [19] in the problem. In [18], Flato et al showed that the nonlinear representation is integrable to a global nonlinear representation of the Poincare group on a dierential manifold, U1 of small initial conditions. This established the existence of global solutions for initial data in U1 at t = 0. They go on to show that the asymptotic representations are also nonlinear and draw conclusions for the infrared tail of the electron. Their results show that \in the classical case (also), one obtains infrared divergencies, if one requires free asymptotic elds as it is needed in QED". In other words, the elds must remain coupled via the self- eld if we are to resolve the infrared problem. In the case where we assume that the system is static and/or stationary (see Section 2), we can make some simpli cations. The system is elliptic in the stationary case (no time dependence). Esteban, Georgiev and Sere showed existence of soliton-like solutions (that is, solutions which are spatially localized) in this case [16]. Furthermore, the wavefunction, together with all its derivatives decreases exponentially at in nity. 2. Some simplified versions of the problem The full 4-dimensional nonlinear problem is somewhat intractable { as stated in Section 1, global existence has only been established recently [18] (and references therein). If we want to get some idea of

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 3

the types of behaviour we might expect in the 4-dimensional problem, we might begin by looking at various subcases. Some simpli ed versions of the problem are as follows: 1. The static case in which we assume that there exists a Lorentz frame in which there is no current \ ow" i.e. j = 0 j0 [29] [4], 2. The stationary case in which we assume (x0; x) = ei!x0 (x) [16], 3. The static spherically symmetric case [29], 4. The static cylindrically symmetric case [4] [8], 5. The static case with z dependence only [6], 6. The 1 + 1 case which was solved exactly for a massless electron by Schwinger in [31], 7. The static axi-symmetric case, 8. The circular current case in which we assume (in spherical coordinates) that j = (j0; 0; j; 0): It appears that the static and stationary assumptions are rather strong, since the resulting system becomes elliptic rather than hyperbolic. In [5] and [30] we proved electric neutrality in the static case (for an isolated system). While this is interesting in that it implies that solutions of this type must consist of an inner charge, say, surrounded by an equal and oppositely charged electron eld i.e. the solutions must be atom-like, it raises the problem of nding a solution that represents a single charged particle. We need to nd a weaker ansatz, perhaps the circular current assumption, with which we have sucient simpli cation without losing the important properties of the 4-dimensional system. It is possible of course, that electric neutrality could be shown to be fully general, which would supporting the conjecture that the total charge of the universe is zero. The larger (quantum cosmological) problem here is the Einstein-Maxwell-Dirac problem, and related problems such as Einstein-Dirac [17] and Einstein-Yang-Mills [3]. Following the methods of the analysts working in relativity theory, our aim in [29] [4] etc was to enumerate the subcases, starting with all possible ODE cases (see Section 4), and then progressing to (static) axi-symmetric and other two-dimensional cases. Meanwhile, the three dimensional static case proved to be somewhat tractable. 3. Using the Clifford Algebra to derive some constraints

In this section, we make use of the properties of the Cliord algebra (which the  represent as 4  4 matrices), to solve the Dirac equation for the potential and to show that there are some useful consisitency

4

HILARY BOOTH

conditions upon the wavefunction and its rst derivatives. We want to write the potential in terms of the wavefunction, so that we can substitute it into the Maxwell equation. A complex Cliord algebra A(n) consists of all possible products of the n basis vectors (of an n-dimensional vector space), ei, which obey the following: (ei )2 = ,1; if i 2 f0; : : : ; n , 1g; (2) eiej + ej ei = 0; if i 6= j: Using the isomorphism [11] (3) A(n + 2) = A(n) A(2) we can construct the representation of A(4), the gamma matrices, from the Pauli matrices which represent A(2). For the construction of higher dimensional Cliord algebras from lower dimensions see for example [14]. One possible representation of A(4) is: 3 3 2 2 0 0 0 1 0 0 0 ,1 7 7 6 6

0 = 64 00 ,10 01 00 75 1 = i 64 00 10 01 00 75 ,1 0 0 0 ,1 0 0 0 2 3 3 2 0 0 ,1 0 1 0 0 0 6 7 7 6 (4) 2 = i 64 00 10 ,10 00 75 3 = i 64 ,10 00 00 ,10 75 0 ,1 0 0 0 0 0 ,1 We note that in this representation, 0 is anti-symmetric (a-s), and i

; i = 1; 2; 3 are symmetric. Whatever representation we use, we can always invert the Dirac equation to express the spatial components of the potential, Ai ; i = 1; 2; 3 in terms of A0 , the wavefunction, and its rst derivatives. Multiplying (1) on the left t i (where t is the transpose of ) we have 1 t i ( @ + im) : t i A (5)  =  ie But the product i j ; i 6= j is (a-s), i i = ,1, and i 0 is symmetric, so that, using the argument following (7) we can write: t A = t 0 i A + 1 t i (  @ + im) : (6) i 0  ie In a similar way (by multiplying the Dirac equation by 0 1 2 3 on the left, and subtracting the conjugate equation premultiplied by

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 5

0 1 2 3) we can solve for A0 . There is a condition to be met here see [29] i.e. that j  is not a null vector which has the physical interpretation that the electron is not travelling faster than the speed of light. In 1957, one of Dirac's students, Eliezer solved for the potential in this way [15]. He went on to show that when we solve for the potential, we must also adopt a consistency condition which applies to . In that paper there was a contribution by Dirac who streamlined some of Eliezer's calculations. Although they used a dierent representation of the algebra, the argument was essentially the following. If , is an antisymmetric (a-s) (4  4) matrix then t , = 0; since this quantity is an a-s scalar. (7) If we can nd a , such that ,  are all a-s, then t , (  A ) = 0;  since A is a scalar, so that if satis es (1) then t , (  @ + im) = 0; (8)  which gives us a consistency condition on and its rst derivatives. The same is true of the complex Dirac equation and we have t (9) , (  @ , im) = 0: We can extend Dirac's argument one step further by premultiplying the Dirac equation by and the complex equation by and noting that since ,  is a-s, t  t ,  A = , t , A : This gives us another condition on namely  t t (10) , ( @ + im) + t , (  @ , im) = 0: There is only one possible element of the Cliord algebra which when premultiplying all of the  yields an a-s matrix. In the representationused here, this is the product , = 1 2 3: Using this in (8) (9) and (10), we have two complex conditions (or four real conditions) on the components of . As is well-known (see [11] for example), even dimensional complex Cliord algebras are simple, that is, they can not be decomposed into the direct sum of two nontrivial subspaces which obey closure under algebraic multiplication. If we want to decompose the 4-dimensional into a 2-dimensional algebra and use 2-spinors, then when must accept

6

HILARY BOOTH

the additional structure in which the two 2-spinor spaces are conjugate dual spaces. See for example [28] in which the 2-spinor formalism is given in terms of In eld-van der Waerden symbols. An argument, equivalent to Dirac/Eliezer's but giving all of the consistency conditions, was given by Radford using the 2-spinor formalism in 1996 [29], and subsequently in [4], although Radford referred to them as \reality conditions", in keeping with the conventions in [28]. We can think of the Dirac equation and its conjugate equations as eight equations in four unknowns (the four real scalars, A ). If we solve for these A then we must have four additional (real) constraints, which correspond to the two complex consistency conditions, con rming that there are no further constraints upon the system. In [7] we outline these conditions and then go on to generalize to higher dimensional cases. Allowing these higher Cliord algebras enables us to pursue the same arguments when applied to supersymmetric matter [7]. See [22] and references therein. 4. ODE solutions Within the static system there are three interesting ODE cases, spherically symmetric, cylindrically symmetric and dependence on z only. The spherical and cylindrical cases were examined extensively in [29] [4]. The case where dependence is on z only, is similar in some respects [5]. We rst apply our consistency conditions to the electromagnetic potential A which has been expressed in terms of Dirac spinors and their rst derivatives (by solving the Dirac equations for the potential as outlined in Section 3). These reality conditions allow us some simpler expressions which are then inserted into the Maxwell equation, resulting in fourth order ODE's. We will also note here that in the 1 + 1 case [12] [13] [6] the system was also reduced to fourth order ODE's, which in some cases are solved explicitly [12] [13], whilst in others we are currently developing more numerical results [6]. When we assume that the Dirac current is static, we lose three (real) degrees of freedom in , since three components of j  are set to zero. See [29] and [4] in which we also x the gauge by choosing 0

(11)

1

= ,Y e 2 , = Xe 2   i

i

(

( + )

)

= ,Y e, 2   = Xe, 2 , ; i

2 3

i

(

( + ) )

where , , X and Y are real functions. These expressions were substituted into the potential (which was solved for in terms of Dirac spinors

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 7

and their rst derivatives), yielding: 2 , Y 2) @ + (r):l (12) A0 = cos  + ((X X 2+ Y 2) @t (X 2 + Y 2 )  @ 1 2 2 (13) A = (X 2 + Y 2) @t l + (X , Y )r , r  l : We now use the consistency conditions, one of which is conservation of charge, which is obeyed automatically as stated in [29]. The other three conditions, in the variables required for the static case, are given below.

@ (X 2 + Y 2 ) = 0 @t

(14) (15) (16)

r:l

= ,(X 2 + Y 2 ) sin  = 0:

@ l + (r)  l @t where l = (2XY cos ; 2XY sin ; X 2 , Y 2).

The Maxwell equations act upon A, as de ned above, and the current vector becomes j  = (2(X 2 + Y 2 ); 0; 0; 0) : We showed in [5] that the static equations, in the gauge given by @X @Y Eq. 11, are stationary if and only if @ @t = 0 and @t = 0 (or @t = 0). In @l the stationary case, @ @t = 0 and @t = 0. Now in the stationary case, the third reality condition Eq. 16 tells us that r is proportional to l and we choose the function  such that l = r sin  r: Substituting this into the expression for the potential Eq. 12 and noting that X 2 + Y 2 = jlj then (17) A = 0

cos   r:l (X + Y 2

s

2 2   = cos   r + 2 + 2 2 : 2) r r sin  2

From here, it is a straightforward calculation to apply symmetry arguments and calculate the resulting ODE's [29] [4] [5]. Then, in dimensionless variables [29] [4], the equations reduce to:

(18)

d dx dF dx dA dx dZ dx

= A , cos  = Z

F f (x) = ,Z sin ; =

8

where

and

HILARY BOOTH

8 2 < x in the spherical case f (x) = : x in the cylindrical case 1 in the z dependent case

8 < r in the spherical case x = :  in the cylindrical case z in the z dependent case, r being the distance from the origin and  the distance from the central

axis. It is easily shown that these four rst order equations can be written as the fourth order,

(19)

d2 f (x)  d2 , sin  d  dx2 dx2 dx   2 d d  d + dx f (x) dx2 , sin  dx sin  = 0:

In the cylindrical case, Z () is the charge per unit ring radius , F () is the charge within a radius , and A is the scalar potential, A0. See [4]. Similar physical quantities are represented in the spherical and z cases. Most importantly, we are looking for solutions whose charge density Z () decreases rapidly towards in nity, so that we can nd solutions which are localized or particle-like. Our zero total charge result [5] tells us to expect that F ! 0 as  ! 1 and likewise, A ! const as  ! 1. As pointed out by Chris Cosgrove (private communication), equation (19) has non-integer resonance numbers [1] [2] and we do not expect to nd an integrable system (in the soliton sense) here. Instead, we show that there are a family of orbits (in the sense of [10]) all of which approach the trivial (constant) solution at in nity. As an example, we look at the cylindrical case, noting that similar results hold in the spherical and z case. The numerical orbits in the Figure 1 complete the results in [4] in which a single member (analytic in 1 ) of these families was shown to exist and calculated numerically. 5. Vacuum Maxwell singularities near the origin In [4] it was shown that in the cylindrical case Lemma 1. Suppose (; F; A; Z ) is a solution to Equation(18) on I = (0; 1), for some 1 ; 0 < 1 < 1. Suppose also that Z  0 is continuous and bounded on I . Then,

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 9

(i) F is C 1 on I and has a well de ned, nite limit as  ! 0. Z has a well-de ned limit as  ! 0. (ii) if F (0) 6= 0 then A is unbounded as  ! 0. In particular, A =

() ln(), where is C 2 and bounded on I, ! F (0) as  ! 0. Also,  is bounded as  ! 0. Lemma 2. Suppose (; F; A; Z ) is a solution to Equation(18) on  2 (0; 1). Suppose also that Z  0 with F continous and bounded on the interval. Then (i) If F (1 )  0 for some 1 2 [0; 1), then  ! 1, A ! 1, and F ! 1 as  ! 1. (ii) If F < 0 on (0; 1) then F ! 0 as  ! 1. In addition, if A and Z have well-de ned limits as  ! 1 then Z ! 0 and A ! A1 as  ! 1, with ,1  A1  1. Similar results were established for the spherical case in [29]. In [29][4], Radford and the current author have shown that the behaviour of solutions near the origin resembles the vacuum solutions of the Maxwell equations. Given that the Maxwell equation can be written as the square of a Dirac operator (see for example [26]), it should be possible to formulate these results, together with exponential decrease in terms of the properties of k-monogenic functions | those functions which are solutions to Dk = D + k0 = 0; where 0 is a basis vector (corresponding to the time coordinate) of a complex Cliord algebra. The fundamental solution of the k-Dirac operator has the same singularity at the origin and decreases exponentially as x ! 1. In this case, the operator has coecients in 1, the subspace of vectors  . We are currently working on a clari cation of this point. 6. Numerical Solutions In [29][4] [8] we showed examples of numerical solutions exhibiting the characteristics referred to in Section 5. Earlier attempts at nding numerical solutions[32] [25] were marred by a \simplifying assumption" which proved to be invalid { solutions did not exist in that case. See [29]. In [8] we noted that there are families of orbits parameterized by the constant cjxj where we assume that solutions are of the form e,cjxjg(jxj), with g(jxj) an analytic function. By varying the value of c, the boundary conditions are perturbed to neighboring solutions

10

HILARY BOOTH

(orbits), all decreasing exponentially at in nity. See Figure 1. All values of c yield solutions which satisfy the two Lemmas, and the zero total charge result. That is, all solutions surround a central wire along the axis of symmetry. Similar numerical solutions can be found in the spherical case [6]. The spherical solutions surround a central Coulomb eld, but the static condition forces a monopole at the origin [29] [5] [30], that is, an unbounded A component. Further ODE solutions occur in the z case [5] and 1 + 1 case, neither of which have been fully examined in the previous literature [12] [13]. These results will be forthcoming also in [6]. Similar spherically-symmetric solutions were found in [27]. This time the Schrodinger-Newton equations provided an identical coupling as in the static Maxwell-Dirac case, which is essentially an elliptic system. More work must be done to investigate the stability of both of these systems. In [27] implications have been developed in the context of quantum gravity, but work in this area is far from complete. We are currently considering the problem in this context. The solutions found in [27] blow-up at larger distances from the origin. However, in the Maxwell-Dirac case, this behaviour appeared only as a numerical anomaly. When the total charge became slightly positive (due to the step-size of the numerical solution), we entered a regime described in Lemma 2 of [4] in which all solutions become unbounded as  ! 1. These solutions were illustrated numerically in [4], but discarded as being of less interest than the bounded solutions. The solution shown in Figure 1 was calculated using the MATLAB ODE solver ODE113. The relative error tolerance was set at 1e , 4 and the absolute error tolerance at 1e , 8. The same behaviour was observed when the tolerances were decreased to 1e , 6 and 1e , 12 or increased to 1e , 3 and 1e , 6. All solutions remained stable whether calculated as a function of increasing or decreasing radius (i.e. shooting away from or towards the central axis). The two dimensional cases (static axi-symmetric, circular current axi-symmetric, the massive 1 + 1 case) still require reliable numerical results. Those available to date [32] [25] have been awed by the imposition of \approximations" which were shown in [29] to be possible only in trivial cases (in which the equations are no longer coupled). 7. Vanishing Total Charge In the solutions in Figure 1 the variable F () representing the total charge within a ring, radius , tends towards zero at 1. Lemma 1 also states that the potential must tend towards a solution which is

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 11 Orbits of charge on ring radius r

chi vs r 0

18 −50

16

14 −100 12

chi

Z

−150 10

8

−200

6 −250 4 −300

2

0

0

5

10

15

0

5

r

10

15

10

15

r Total charge within radius r

Potential vs log r 0

700 −10

600

−20

−30

500

−40

A

F

400 −50

300 −60

200

−70

−80

100

−90

0 −1 10

0

10

1

10

0

5 r

Figure 1. A family of orbits exhibiting the properties of Lemma 1 and Lemma2.

logarithmic in , near the central axis. This corresponds to a central charged wire (along the z axis) which is a solution to the vacuum Maxwell equations. As such, we can think of this part of the solution as representing an external eld. (The scalar potential could be separated at this point into Aexternal + Afermion+interaction , since the Aexternal , as the homogenous solution, contibutes nothing to the coupling between the Maxwell and Dirac equations.) Physically, this means that the (inner) external eld must be surrounded by an equal and oppositely charged eld. The total electric charge of the system Z , 1 0
:dS ; with S the unit ring of radius ; lim r A  !1 4 S must vanish.

12

HILARY BOOTH

In [5], we were able to show that this is the case for all stationary solutions, given that they are in a \reasonable" function space. If we de ne an isolated system as one for which all sources are contained in some ball Bk (k < 1) and for which the elds die o as jxj = r ! 1, then we showed in [5] and [30] that an isolated, stationary, static Maxwell-Dirac system is electrically neutral. A similar result, for the total electric charge per unit length in the z direction (or unit ring), holds in the cylindrical case. This shows mathematically, that a (stationary) solution must be atom-like (in the sense that any central charge must be surrounded by an equal and opposite charge. In addition it shows that there cannot be a stationary solution representing an isolated electron (or even one that rotates around the axis with constant velocity). Note also, that the signs of the charges can be reversed, giving us negatively charged singularites and positively charged \electron" elds. All results remain unchanged under such a reversal of sign. The total charge must vanish overall. A more dicult problem presents itself if we are to allow for two fermion elds of opposite charge, for example, the solution corresponding to positronium. We are currently considering ways in which we can have have solutions of opposite charge interacting together. 8. Discussion and Conclusions Until the mathematical problems of QED have been resolved, all methods of addressing the equations governing the interaction between electrons and elds are signi cant. A non-perturbative QED must consist of classical and/or semi-classical arguments which aim to justify the Feynman integral method, and the quantized eld approach. For the coupled M-D system, we have existence (for small initial data), existence for the stationary case, and descriptions of various subcases. But analysis of the system is far from complete. We have indicated, within this review paper, the directions we are currently pursuing.

References

[1] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution equations and ordinary dierential equations of P-type.I J.Math.Phys. 21 (4), 715-721 (1980) [2] M.J.Ablowitz,A.Ramani,H.Segur A connection between nonlinear evolution equations and ordinary dierential equations of P-type.II J.Math.Phys. 21 (5), 1006-1018 (1980)

ELECTRONS WITH SELF-FIELD AS SOLUTIONS TO NONLINEAR PDE 13 [3] R.Bartnik, J McKinnon, Particlelike solutions of the Einstein-Yang-Mills equations Phys. Rev. Lett., 61 141-144, 1988. [4] H.S.Booth, C.J.Radford, The Dirac-Maxwell equations with cylindrical symmetry J. Math. Phys. 38 (3), 1257-1268 (1997). [5] H.S.Booth, The Static Maxwell-Dirac Equations PhD Thesis, University of New England (1998). [6] H.S.Booth Various ODE Solutions to the static and 1+1 Maxwell-Dirac Equations (in preparation). [7] H.S.Booth,P.Jarvis,G.Legg, Algebraic solution for the vector potential in the Dirac equation (in preparation). [8] H.S.Booth, Nonlinear electron solutions and their characteristics at in nity The ANZIAM Journal (formerly J.Aust.M.S (B)) (to appear). [9] J.Chadam Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension J.Funct.Anal. 13, 173-184 (1973). [10] K.Cieliebak and E.Sere, Pseudoholomorphic curves and multiplicity of homoclinic orbits, Duke Mathematical Journal 77, (2) 483-518 (1995). [11] R.Coquereaux, Modulo 8 periodicity of real Cliord algebras and particle physics, Phys Lett 115B (5), 389-395 (1982). [12] A.Das, D.Kay, A class of exact plane wave solutions of the Maxwell-Dirac equations, J.Maths.Phys. 30 (10) (1989) 2280-2284. [13] A.Das, General Solutions of Maxwell-Dirac equations in 1+1 dimensional space-time and a spatially con ned solution, J.Maths.Phys. 34 (9) (1993) 39863999. [14] R.Delanghe,F.Sommen,V.Soucek, Cliord Algebra and Spinor-Valued Functions, KLUWER (1992). [15] C.J. Eliezer, A Consistency Condition for Electron Wave Functions, Camb.Philos. 54, 2 (1957). [16] M.Esteban, V.Georgiev, E. Sere, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. 4, 265-281, (1996). [17] Felix Finster, Niky Kamran, Joel Smoller, Shing-Tung Yau, Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry CPAM 53,7, 902-929 (2000). [18] M. Flato, J.C.H. Simon, E Ta in, Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations Memoirs of the American Mathematical Society, (May 1997). [19] V.Georgiev, Small amplitude solutions of the Maxwell-Dirac equations Indiana Univ. Math. J. 40 (3), 845-883 (1991). [20] R.T.Glassey, W.A.Strauss Conservation Laws for the Maxwell-Dirac and Klein-Gordon-Dirac equations, J.Math.Phys., 20, (1979). [21] L.Gross The Cauchy problem for the coupled Maxwell and Dirac equations Comm. Pure Appl. Math. 19 1-5 (1966). [22] P.S.Howe, G.Sierra, P.K.Townsend Supersymmetry in six dimensions Nuclear Phys B221 331-348 (1983). [23] E.H. Lieb and M. Loss, Self-Energy of Electrons in Non-perturbative QED , in Dierential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, R. Weikard and G. Weinstein, eds. 255-269 Internat. Press (1999). arXiv math-ph/9908020, mp arc 99-305.

14

HILARY BOOTH

[24] E.H. Lieb and M. Loss, Remarks about the ultraviolet problem in QED, (in preparation). [25] A.G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J.Phys.A 28 No.18, 5385-5392 (1995). [26] A.McIntosh, Cliord algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains in: Cliord Algebras in Analysis and Related Topics (John Ryan ed), 33-88 CRC Press Boca Raton, FL, 1996. [27] I.M.Moroz,R.Penrose,P.Tod, Spherically-symmetric solutions of the Schrodinger-Newton equations, Class. Quantum Grav. 15 2733-2742 (1998). [28] R.Penrose,W.Rindler, Spinors and Space-Time I (Two Spinor Calculus and Relativistic Fields), CAMBRIDGE UNIVERSITY PRESS, (1984). [29] C.J. Radford, Localised Solutions of the Dirac-Maxwell Equations, J. Math. Phys. 37 (9), 4418-4433 (1996). [30] C.J.Radford, H.S.Booth Magnetic Monopoles, electric neutrality and the static Maxwell-Dirac equations J.Phys.A:Math.Gen. 32, 5807-5822 (1999). [31] J.Schwinger Gauge Invariance and Mass. II Phys. Rev. 128 (5), 2425-2429 (1962). [32] M. Wakano, Intensely Localized Solutions of the Dirac-Maxwell Field Equations, Prog.T.Phys. 35, 1117-1141, (1966). Centre for Mathematics and it Applications, Australian National University, Canberra ACT 0200 Australia

E-mail address :

[email protected]

QUANTUM MECHANICS AS AN INTUITIONISTIC FORM OF CLASSICAL MECHANICS JOHN V. CORBETT AND MURRAY ADELMAN

Abstract. Intuitionistic real numbers are constructed as sheaves

on the state space of the Schrodinger representation of a CCRalgebra with a nite number of degrees of freedom. These numbers are used as the values of position and momentum variables that obey Newton's equations of motion. Heisenberg's operator equations of motion are shown to give rise to numerical equations that, on a family of open subsets of state space, are local approximations to Newton's equations of motion for the intuitionistically valued variables.

Introduction

Do sub-atomic particles have positions and momenta at all times? Can the numerical value of the position of such a particle always be given a triplet of real numbers? What real numbers should be used? In this paper we argue that there is a class of real numbers that can be used to label the positions and momenta of sub-atomic particles at all instances of time. Furthermore if these numbers, as values of positions and momenta, are assumed to satisfy the equations of motion of classical mechanics, then these classical equations are approximated locally by the quantum mechanical operator equations of motion, restricted to act on certain subsets of state space. The real numbers that we use are real numbers in the topos of sheaves on the state space of the Schrodinger representation of a CCR-algebra with a nite number of degrees of freedom. In the standard Hilbert space framework for quantum mechanics[1] the algebra of the canonical commutation relations (CCR) is generated by the operators Qj = Qj , Pj = Pj , j = 1; : : : ; n, and the identity operator I which satisfy the relations [Pj ; Qj ] = ,iI , [Pj ; Qk ] = 0 if k is dierent from j , [Qj ; Qk ] = 0, [Pj ; Pk ] = 0 for all j; k. We have put Planck's constant divided by 2 equal to one. We will further simplify the notation by taking n = 1. 15

16

JOHN V. CORBETT AND MURRAY ADELMAN

1. The Schrodinger representation The Schrodinger representation of the CCR-algebra M is the representation in which the Hilbert space is L2 (R). Q is represented by multiplication by the real variable x and P by (1=i) times the operator of dierentiation with respect to x. Let S (R ) denote the Schwartz space of in nitely dierentiable functions of rapid decrease on R . Then the physical quantities are represented by self-adjoint  is the elements in the closure M of the CCR-algebra M , where M  , smallest closed extension of M . M = X X 2 M and X , is the restriction to S (R ) of the Hilbert space closure of X . Following the de nitions of Powers[7], M is essentially self-adjoint because the adjoint M  of M equals the closure M of M . 2. The State Space ES We give only a resume of the results, for more details see Inoue[8]. A linear functional f on M is strongly positive i f (X )  0 for all X  0 in M . De nition 1. The states on M are the strongly positive linear functionals on M that are normalised to take the value 1 on the element I of M . Theorem 1 (Inoue[8]). Every strongly positive linear functional on M is given by a trace functional. De nition 2. The state space ES of the Schrodinger representation of M is the set of all strongly positive linear functionals on M that are normalised. The state space ES is contained in the convex hull of projections P onto one-dimensional subspaces spanned by unit vectors u 2 S (R 3 ). Recall that each state  2 ES is a trace class positive bounded operator P with trace 1, which can be written as  = PnPn, where the sum over n may go from 1 to 1. For all n, n  0, n = 1 and the Pn are orthogonal projections. The following result demonstrates that all states in ES must satisfy the further condition that as n approaches in nity the sequence fng converges to zero faster than any power of 1=n. Theorem 2. Given  = P nPn 2 ES , where the projections Pn project onto one-dimensional subspaces spanned by unit vectors un 2 S (R 3 ). tr A is nite for any self-adjoint operator A in thekSchrodinger representation of the CCR-algebra M if and only if lim n n = 0, for all integers k > 0.

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

17

Let A be a self-adjoint operator in M and  is a state on M which belongs to ES . Then for any orthonormal basis of P H, fumg, m = 1 to 1, which is composedPfrom elements in S , tr A = (um ; Aum) . Now using  = nPn the trace becomes a double sum X X,
um ; nPnAum : tr A = Because the trace is independent of the basis used to calculate it, we can choose the ortho-normal basis fumg to be in the ranges of the Pn so that the double sum reduces to X
, tr A = m um ; Aum : Since A is in the algebra M , it is a polynomial of some nite degree k in the self-adjoint operators P , Q. By Lemma 18 in Jae[10], A is majorized by a polynomial of degree k in the self-adjoint operator P H = (P + Q ), ,
,
um ; Aum  um ; pk (H )um : When the fumg are the eigenfunctions of H , pk (H )um = pk (m)um, where m = (2m + 1) are the eigenvalues of H , so that the absolute P value of the m term of the series m (um; Aum) satis es 0  m (um; Aum) ,
1  m um; H (m)um 2 = m pk (m ) 21 : Therefore if lim nk n = 0, for all k > 0, then 9 a positive integer N and a positive constant C such that, for all m > N , mpk (m) 21 < mC P and so the series m (um; Aum) converges absolutely by the comparison test. On the otherPhand, if tr A is nite for all self-adjoint operators A then the series m (um; Aum) must converge. If A is positive then the convergence is absolute so that when A is a monomial of degree k in the Hamiltonian operator H ,then lim m(m) 21 k = 0 for all positive k which shows the converse is true. The same results hold even when the fumg are not the eigenfunctions of H . The minimax principle for eigenvalues of symmetric operators, see for example, Kato[6] section I.6.10, implies that ,
um; pk (H )um  pk (m) ,
because it states that the maximum of v; pk (H )v , when v is a unit vector and (v; ei) = 0, i = 1;


; n , 1, for any orthonormal set of vectors fei g, is the eigenvalue pk (m) obtained when the feig are the Proof.

1 2

2

2

2

1 2

th

2

( )

18

JOHN V. CORBETT AND MURRAY ADELMAN

rst n , 1 eigenvalues of H . This result is applicable here to the set of vectors fuig, i = 1;


; m, with v = um. We have shown that tr A is nite for any self-adjoint operator A in this representation of the CCR-algebra M if and only if lim nk n = 0, for all integers k > 0. The topology on E is chosen to make the functions A^, where A^() = tr A, continuous when A is an essentially self-adjoint continuous operator on S (R ). S

3. The topos Shv(E ) A sheaf Y on a topological space X can be described[3] by a rule which assigns to each point x of X a set Y (x) consisting of the germs of a prescribed class of functions, where the germs of the functions are de ned in neighborhoods of the point x. The collection of sets Y (x) which are labelled by points x in X can be glued together to form a space Y in such a way that the projection from Y onto X is a local homeomorphism; that is, for each x in X and each y on the ber above x (i.e. for each such y the projection of y onto X is x) there is a neighborhood N of y such that the projection of N onto X is a neighborhood of x. A section of the sheaf Y over the open subset U of X is a function s from U to Y that belongs to the prescribed class of functions and satis es the condition that, for all x in U , the projection of s(x) onto X is x. The sheaf construction allows a section f de ned on the open set U to be restricted to sections f V on open subsets V contained in the open set U and, conversely, the section f on U can be recovered by patching together the sections f V where V belongs to an open cover of U . A spatial topos is a category of sheaves on a topological space. The objects of this category are sheaves over the topological space and the arrows are sheaf morphisms, that is, an arrow is a continuous function that maps a sheaf Y to a sheaf Y in such a way that it sends bers in Y to bers in Y , equivalently, sections of Y over U to sections of Y over U , where U is an open subset of X . The topos Shv(E ) of sheaves on the topological space E is constructed in this way. In 1970, Lawvere[2] showed that toposes can be viewed as a \variable" set theory whose internal logic is intuitionistic. The propositional calculus of the logic of the spatial topos of sheaves over X is the Heyting Algebra[3] of the open subsets of X . This means that as well as S

0

0

0

0

0

S

S

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

19

being true or false, propositions in this logic can be true to intermediate extents which are given by open subsets of X . True corresponds to the whole set X . False corresponds to the empty set. There exist Boolean algebras in which propositions can be true to varying extents but, in addition, the Heyting algebra of open sets does not satisfy all the laws of classical logic. Two of the most striking dierences between classical and intuitionistic logics are that the law of the excluded middle and the Axiom of Choice do not hold for intuitionistic logic. We have argued elsewhere[5] that aspects of quantum mechanics, such as the two slit experiment,that are dicult to understand with Boolean logic are better described using intuitionistic logic. There is an analogy between the language of toposes and that of sets which makes it easier to work in toposes. Sheaves in a topos correspond to sets, subsheaves of a sheaf to subsets and local sections to elements of a set. Then as long as a proof in set theory does not use the law of excluded middle or the Axiom of Choice then it can be translated into a proof in topos theory.

4. Real Numbers in Spatial Toposes Dedekind numbers are de ned to be the completion of the rational numbers obtained by using cuts, and Cauchy numbers are de ned as the completion of the rationals obtained by using Cauchy sequences. These dierent constructions can only be shown to be equivalent by using either the Axiom of Choice or the law of the excluded middle.[3] Therefore, when intuitionistic logic is assumed, we expect that these two types of real numbers are not equivalent. It has been shown[4] that in a spatial topos the sheaf of rational numbers Q is the sheaf whose sections over an open set U are given by locally constant functions from U with values in the standard rationals while the sheaf of Cauchy reals R C is the sheaf whose sections over an open set U are given by locally constant functions from U with values in the standard reals. A function is locally constant if it is constant on each connected open subset of its domain. On the other hand the sheaf of Dedekind reals R D is the sheaf whose sections over U are given by continuous functions from U to the standard reals. The Cauchy reals form a proper sub-sheaf of the Dedekind reals unless the underlying topological space X is the one point space.

20

JOHN V. CORBETT AND MURRAY ADELMAN

5. The Quantum Reals By the construction of the topology on ES for any self-adjoint operator A in the Schrodinger representation of the CCR, the function tr A is a globally de ned continuous function and therefore in R D . We interpret the functions A() = tr A, with domains given by open subsets of state space, to be the numerical values of the physical quantity that is represented by the self-adjoint operator A. Not every Dedekind real number is of this form. Real numbers of this form are a proper subsheaf of the Dedekind real numbers in the spatial topos of sheaves on the state space ES . We will call them the quantum reals, they belong to the sheaf of locally linear functions.[5] De nition 3. If U is an open subset of ES then the function f from U to the standard reals is locally linear at  in U and there is an open neighborhood U 0 of , with U 0 inside U , and a bounded self-adjoint operator A such that f U = A^ U . De nition 4. The sheaf of locally linear functions, A , is de ned by its sections over any open subset U of ES as the set of all locally linear functions on U with the requirement that if the open set V is contained in U then the sheaf of locally linear functions over V is obtained by restricting the locally linear functions over U . The global elements of the locally linear functions are given by the functions A^, where A is a self-adjoint operator, that are continuous on S . It suces to de ne algebraic relations between elements of A globally, because ES is locally connected and so we can treat functions which are de ned on disjoint connected components as if they were globally de ned. When U is an open neighbourhood of the state  then the quantum real numbers belonging to the sections of A over U can be thought of as real numbers tangent to those Dedekind reals that have a tangent space at . 6. The Dedekind Reals R D Stout[4] has shown that the usual order on the rational numbers Q can be extended to the following order on R D . De nition 5. The order relation < on the Dedekind reals, R D , is given by the de nition: ,
x < y if and only if 9 q 2 Q (q 2 x+ ) ^ (q 2 y,) where x+ is the upper cut of x and y, is the lower cut of y . The relation < is the subobject of R D R D consisting of such pairs (x; y). 0

0

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

21

Trichotomy does not hold universally for the order < on R D . The order  on R D has the property that x  y is not the same as (x < y) _ (x = y). De nition 6. The order relation x  y is the subobject of R D R D consisting of the pair (x; y) with x+  y + and y,  x, , where x+ is the upper cut of x and x, is the lower cut of x, and similarly for y . Stout[4] also showed that the statement (x  y) is equivalent to the statement :(y < x). De nition 7. The open interval (x; y) for x < y is the subobject of R D consisting of those z in R D that satisfy x < z < y . The closed interval [x; y ] for x < y is the subobject of R D consisting of those z in R D that satisfy x  z  y. The open intervals can be used to construct an interval topology T on R D analogously to the interval topology on the standard real numbers that is generated from the open intervals by nite intersection and arbitrary union. The topology T on R D is such that Q is dense in R D with respect to T. If the max function is de ned using the order  by the conditions: (i) x  max(x; y) and y  max(x; y), and (ii) if z  x and z  y then z  max(x; y) and the norm function: j j : R D ! R D is de ned by jxj = max(x; ,x), then the norm j j satis es the usual conditions of non-negativity, that only 0 has norm zero and that the triangle inequality holds.

(R D ; T ) is a metric space with the metric d(x; y) = x , y . It is both complete and separable[4]. R D is a eld in the sense that for all a in R D ,if a does not belong to the sheaf of germs of invertible functions, Unit(R D ),then a = 0. 7. Properties of the Quantum Reals Theorem 3. A is a proper sub-sheaf of the sheaf of Dedekind numbers R D and is dense in R D in the metric topology T . The sheaf A inherits the orders  and < from R D . On the other hand A can be ordered as a consequence of the orders on the selfadjoint operators: 1. A is strictly positive, A > 0, if (Au; u) > 0 for u 6= 0, u 2 D(A). 2. A is non-negative, A  0, if (Au; u)  0 for all u 2 D(A). Lemma 8. The orders  and < on A inherited from R D are equivalent to those obtained from those on continuous self-adjoint operators.

22

JOHN V. CORBETT AND MURRAY ADELMAN

Proof. A is a non-negative self-adjoint operator i tr A  0 for all  in ES , i.e. A^  0 globally. When, as a Dedekind real number, a = A^  0 then 0+  a+ and a,  0,. Globally, 0+ = fq 2 Q j q > 0g and 0, = fq 2 Q j q > 0g so that, if a = A^  0 then A is a non-negative operator and if A is a non-negative operator then a = A^  0. The positivity order for a continuous self-adjoint operator A is equivalent to A being bounded away from zero, i.e. there exists a rational number q > 0 such that (u; Au) > q for all u 6= 0. This gives the equivalence of the operator > with the > for Dedekind reals restricted to A , because for the latter a > 0 means globally that 9q 2 Q with q 2 0+ and q 2 a,. If a = A^ and b = B^ then the R D distance between a and b is given by the metric ja , bj on RD . There is another metric on the quantum numbers A ,1 it is given by the number jA\ , B j, where jC j is the operator jC j = (C 2) 2 , when C is self-adjoint. Proposition 9. The two metrics coincide on A , that is, jA\ , Bj = ^ ^ ja , bj, for all pairs of quantum numbers a = A and b = B . Proof. It is sucient to let B = 0. We will only consider the global sections. see for example section VI.2.7 of Kato[6], that It is well-known, (u; Au)  (u; jAju) for all u in the domain D (A) = D (jAj), whence tr A  tr jAj for all  in ES , i.e. jA ^j  jc Aj. ^ ,A^) and jc Aj = jc Aj. The As on elements of R D , jA^j = max(A; ^ ^ ^ lower cut of jAj is the union of the lower cuts of A ,and
of ,A, that is jA^j, = (A^), [ (,A^),, which means that jA^j,  jc Aj ,. The upper cut of jA^j is the intersection of the upper cuts of A^ and of ,A^, that is q 2 Q belongs to, jA^j
+ if q is greater than or equal to both A^ and of ,
+ + ^j+, therefore jA^j+  jc ,A^. Thus if q 2 jc Aj then q 2 j A A j . c c ^ ^ This shows that jAj  A and therefore jAj = A . The metric is used to de ne Cauchy sequences in R D , this result means that we can de ne Cauchy Sequences in A in the same way. In order to ensure uniqueness of the limits of a Cauchy sequence we need a concept of apartness, >< b, i (a > b) _ (a < b). Proposition 11. a >< b i ja , bj > 0.

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

23

8. Calculus of Functions The concept of limit is available for functions: R D ! R D and dierential calculus for such functions can be developed. The de nition of the limit of a function G 2 R D is modeled on the standard de nition; the limit of G as x tends to b is L i 8n > 0; 9 m > 0; 8x 2 R D ; 0 < jx , bj < 1=m =) G(x) , L < 1=n . In this de nition both m and n are in Q . The uniqueness of the limit L holds in the sense that if L and M are two limits of G as x tends to b then L and M are apart, L , M = 0. For the requirements of this paper it is enough to consider only polynomial functions. For any b 2 R D all powers bn of b are in R D for n a natural number because products of continuous functions are continuous. Let both b and c 2 R D , then bc = cb 2 R D for the same reason. The product, however, is de ned only on the intersection of the domains of the continuous functions so that care has to be taken with the extent to which the product exists. Sums of numbers b + c exist to the extents given by the intersection of the extents of b and c. Hence we can construct polynomials

F (b) =

X

am bm

where the sum is over nitely many terms and the coecients am 2

R D . We can construct power series by de ning the convergence of the sequence of partial sums in the metric on R D .

The derivative of bm = mbm,1 which implies that the derivative of a monomial is de ned to the same extent as the monomial. This allows us to obtain the derivatives of polynomials and power series. We de ne continuity of a function F : R D ! R D at b in its domain by the requirement that (1) 8n > 0; 9m > 0; 8x 2 R D ;

0 < jx , bj < m1 =) F (x) , F (b) < n1 :

In this de nition we have taken m and n to be in Q . A function is continuous on an interval I in R D i it is continuous at each number in I . Interesting phenomena occur in this calculus and they deserve more study. However we have enough structure now to return to the dynamics of quantum mechanics.

24

JOHN V. CORBETT AND MURRAY ADELMAN

9. Comparing the Dynamics Consider the example of a non-relativistic quantum particle of positive mass  that moves in a central force eld which is derived from a potential function V . We assume that the quantum values X^ of the position of the particle satisfy Newton's equations of motion globally, that is, DQ^ = 1 P^ (2)



and (3) DP^ = F (Q^ ) or (4)  D2(Q^ ) = F (Q^ ) where P^ represents the momentum of the particle and D denotes dierentiation with respect to time. F represents the force, it is the negative gradient of the potential function V as a function: R D ! R D . We will now prove that standard quantum mechanics is a local approximation to a global classical mechanics. More precisely, the theorem states that if the quantum values of the components of position and momentum of a particle are assumed to satisfy Newton's equations of motion globally then the self-adjoint operators corresponding to these values locally satisfy equations that well approximate Heisenberg's equations of motion. The theorem relates a set of operator equations (Heisenberg's equations) to a set of numerical equations (Newton's equations). The straightforward way to get a numerical equation from an operator equation is to multiply each side of the operator equation by a suitable trace class operator (a state) and then take the trace of each side. The original operator equation has become a family of numerical equations which is labelled by the states. Recall that Heisenberg's equations for an operator A are (5) DA = ,i[A; H ] where H is the Hamiltonian operator of the system and the square bracket denotes the operator commutator. Therefore, we get Heisenberg's equations of motion, when Q and P represent operators and the Hamiltonian operator is H = (1=2)P 2 + V (Q), to be DQ = 1 P (6)



QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

25

and (7) DP = i[P; H ] = F (Q) : If we multiply each side of equations (6) and (7) by the operator  and then take the trace of each side we get Heisenberg's numerical equations: D(tr Q) = 1 tr P (8) 

and ,
,
(9) D(tr P ) = tr (DP ) = tr  i[P; H ] = tr  F (Q) : Equation (8) is just Newton's equation (2) at the state . Equation (9) is similar to Newton's equation (3) at the state . The dierence between equations (3) and (9) is the same as that which restricts the validity of the result known as Ehrenfest's Theorem, namely, that, in general, (10) tr F (Q) = F (tr Q): To simplify the discussion we remove the explicit dependence of the equations (2), (3), (8), and (9) on the operators P and use equations of motion in the form of the second order dierential equations: Newton's equations to the extent W are ,
,
(11) D2 Q^ W = F Q^ W : Heisenberg's numerical equations to the extent W are ,
(12) D2 Q^ W = F (Q) W : It is possible, however, that the dierence between the two sides of (10) is small at some state a and remains small for all states in an open neighborhood of a . In this case the equations (11) and (12) are approximately the same in that open set. We deduce that Heisenberg's equations approximate Newton's equation in that open set because if (11) holds in an open set W then it holds at each  in W . We claim that for a suitable class of functions F , Heisenberg's equations approximate Newton's equations locally. The localness of the assertion is twofold, we mean that for every standard real number r we ^ can nd an open set,W , on which both the number Q W is arbitrarily ,
close to r, and F (Q) W is arbitrarily close to F Q^ W . The physical interpretation is that if an observer's measurement apparatus is located in a neighbourhood of the position r (in this world with one spatial dimension) then the dierence between the accelerations of the particle due to the two forces cannot be distinguished with this apparatus. 6

[

[

26

JOHN V. CORBETT AND MURRAY ADELMAN

The class of suitable functions is de ned through the concept of S continuity. De nition 12. We call functions G, S -continuous, if they are realvalued continuous functions on R such that,for the position operator Q, G(Q) : S ! S , is continuous in the standard countably normed topology on the Schwartz space S . Theorem 4. If the force F is S -continuous, then given " > 0, Heisenberg's equations of motion approximate Newton's equations of motion to within " on each member of a collection of open sets fW (r; ")g of state space ES , indexed by the standard real numbers r and " > 0. That is, on each fW (r; ")g, ,
[ ^ W < " : (13) F (Q) W , F Q Proof. The idea behind the proof is to nd states r on which F (tr r Q) closely approximates tr 
r F (Q), then F (Q^ ) will be close to F[ (Q) when , [ ^ ^ ^ ^ F (Q) is close to F Q(r ) 11 and F (Q) is close to tr r F (Q)11 as Dedekind numbers. The following pair of lemmas complete the proof. Lemma 13. If Q is a self-adjoint operator which has only absolutely continuous spectrum, then for any real number r in its spectrum we can construct a sequence of pure states fng such that for the given S -continuous function F , lim tr nF (Q) = F (r) : Proof. Weyl's criterion for the spectrum of self-adjoint operators[9] implies that, for any number r in the spectrum of Q, there exists a sequence of unit vectors fung, in the domain of Q, such that

lim (Q , r)un = 0 : Take n to be the projection onto the one dimensional subspace spanned by un. It is easy to check that lim tr nQ = r : The vectors fung can be chosen to be in S . For example, for any positive integer n, let 1

1

un(x) = n 2  , 4 exp

,

, 21 n2(x , r)2




The sequence fung satis es the requirements of Weyl's lemma for the operator Q and the number r.

QUANTUM MECHANICS AS INTUITIONISTIC MECHANICS

27

f g S

Furthermore we can nd a sequence of vectors un in so that for n large enough the support of un lies in a narrow interval centred on r . Then, by the spectral theorem for Q, the corresponding pure states n form a sequence so that both

f g

lim tr n F (Q) = F (r)

and

lim tr nQ = r :

From Lemma 13, given a real number r in the spectrum of Q, the S continuous function F and a real number " > 0, there exists an integer N such that, for all j > N , both tr j F (Q) , F (r ) < 1 " 6 and (tr j Q) , r <  where  is such that F (r ) , F (x) < 1 " (14) when jr , xj <  : 6 We choose r = j , for some j > N , and deduce that tr r F (Q) , F (tr r Q) < 1 " (15) 3 because tr r F (Q) , F (tr r Q) = tr r F (Q) , F (r) + F (r) , F (tr r Q)  tr r F (Q) , F (r) + F (r) , F (tr r Q) : With this choice of r ,the open set W (r; ") can be de ned as W (r; ") = N (r ; Q;  ) \ N (r ; F (Q); 31 ") where  satis es the requirements (14), and for any S -continuous function F and " > 0 we have that N (r ; F (Q); ") is given by ,




N r ; F (Q); "

Lemma 14. all



(16)

in

When

W (r; "),

r

=

n



 ; tr F (Q)

o

, tr  F (Q) r

is chosen so that equation

tr F (Q)



, F (tr Q) < " :

1 (see e.g. [VSC, xVIII.2]). We call G a homogeneous group if there exists a family of dilations on G. A family of dilations on a Lie group G is a one-parameter group (~t )t>0 (~t  ~s = ~ts ) of automorphisms of G determined by (3) ~t Yj = td Yj ; where Y1; : : : ; Yl is a linear basis of Lie algebra of G and dj  1 for 1  j  l (see [FS]). We say that an operator L de ned by (2) is homogeneous if ~t Xi P = tXi for 1  i  k. For a homogeneous Laplace operator d0 = d1 = lj=1 dj (see [FS]). Spectral multiplier theorems for homogeneous Laplace operators acting on homogeneous groups were investigated by Hulanicki and Stein [HS] (see also [FS, Theorem 6.25]), De Michele and Mauceri [dMM]. The following theorem was obtained independently by Christ [Ch2] and Mauceri and Meda [MM]. See also [Si3]. Its proof relies on heat kernel bounds, L2 estimates from Plancherel theorems, translation and dilation invariant structures of homogeneous groups, Calderon Zygmund operator theory and interpolation theory. Theorem 1. Let L be a homogeneous operator de ned by the formula (2) acting on a homogeneous group G. Denote by d = d0 = d1 homogeneous dimension of the underlying group G. Next suppose that s > d=2 and that F : [0; 1) ! C is a bounded Borel function such that (4) sup k t F kW 2 < 1; j

t>0

s

58

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

where t F () = F (t), kF kWsp = k(I , d2 = dx2 )s=2F kLp and  2 Cc1(R+) is a xed function, not identically zero. Then F (L) is of weak type (1; 1) and bounded on Lq when 1 < q < 1. Note that condition (4) is independent of the choice of  2 Cc1(R+). The Hormander multiplier theorem describes the Fourier multiplier on Rd (see [Ho1]). If we apply Theorem 1 to Rd we obtain a result equivalent to the Hormander multiplier theorem restricted to radial Fourier multipliers. Therefore we call Theorem 1 the Hormander-type multiplier theorem and condition (4) the Hormander-type condition. Theorem 1.1 is optimal for a general homogeneous group, see estimate 1.6. However, for speci c groups such as Heisenberg and related groups, it is possible to obtain multiplier results where the number of derivatives needed is roughly half of the topological dimension n. Often the homogeneous dimension d is strictly greater than the topological dimension n, hence the number of derivatives needed could be less than d=2. See for example, [MS, He4, Du, CS]. In the setting of general Lie groups of polynomial growth spectral multipliers were investigated by Alexopoulos. Note that in this setting, the local dimension d0 and dimension at in nity d1 are dierent in general and the group G does not have dilation invariants as in the case of a homogeneous group. Using nite propagation speed property, estimates on upper bounds of heat kernels and their space gradients, Alexopoulos proved the following (see [Ale1]). Theorem 2. Let L be a group invariant operator acting on a Lie group of polynomial growth de ned by (2). Suppose that s > d=2 = max(d0; d1)=2 and that F : [0; 1) ! C is a bounded Borel function such that (5) sup k t F kWs1 < 1; t>0

where t F () = F (t) and kF kWsp = k(I , d2 = dx2 )s=2F kLp . Then F (L) is of weak type (1; 1) and bounded on Lq when 1 < q < 1. Condition (5) is also independent of the choice of . We note that Theorem 1.2 does not appear exactly the same but is essentially equivalent to the results of [Ale1]. In [He3] Hebisch extended Theorem 2 to a class of abstract operators acting on spaces satisfying \doubling condition" (see also [Ale2]). The order of dierentiability in the Alexopoulos-Hebisch multiplier theorem is optimal. This means that for any s < d=2 we can nd a function F such that F satis es condition (5) but F (A) is not of weak type (1; 1). Indeed, let A be a uniformly elliptic, self-adjoint second-order

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS

59

dierential operator on Rd, e.g. A =
d , where
d is the standard Laplace operator. One can prove that (6) C1(1 + jj)d=2  kAikL1 !L1 1  C2(1 + jj)d=2 (see [SW]). (See also [St1, pp. 52] and Christ [Ch2]). However, if we put F () = jji, then C10 (1 + jj)s=2  sup ktF kW 1  C20 (1 + jj)s=2: ;

t>0

s

Therefore for any s < d=2 Theorem 2 does not hold. If X is a manifold with exponential volume growth, i.e. V (x; r)  cekr and L is the Laplace-Beltrami operator, spectral multipliers was investigate by M. Taylor in [Tay] where it was shown that a sucient p condition is that F is holomorphic on a strip of width k for F ( L) to be bounded on Lp for 1 < p < 1. For more speci c spaces such as certain Iwasawa AN groups, see [He5, CGHM] where it was shown that only a nite number of derivatives are required for F (L) to be bounded on L1 or to be of weak type (1,1). Note that the exponential volume growth is like the dimension d = 1. The theory of spectral multipliers is related to and motivated by the study of convergence of the Riesz means or convergence of other eigenfunction expansions of self-adjoint operators. To de ne the Riesz means of the operator A we put  ) for   R  (7) R () = (1 , =R 0 for  > R: We then de ne the operator R (A) using (1). We call R (A) the Riesz or the Bochner-Riesz means of order . The basic question in the theory of Riesz means is to establish the critical exponent for the continuity and convergence of the Riesz means. More precisely we want to study the optimal range of  for which the Riesz means R (A) are uniformly bounded on L1 (X ) (or other Lq (X ) spaces). Since the publication of Riesz paper [Ri] the summability of the Riesz means has been one of the most fundamental problems in Harmonic Analysis (see e.g. [St2, IX.2 and xIX.6B]). Despite the fact that the Riesz means have been extensively studied we do not have the full description of the optimal range of  even if we study only the space L1(X ). On one hand we know that for the Laplace operator
d = P d 2 d k=1 @k acting on R and the Laplace-Beltrami operator acting on compact d-dimensional Riemannian manifolds the critical exponent is equal (d , 1)=2 (see [So1]). This means that Riesz means are uniformly continuous on L1 (X ) if and only if  > (d , 1)=2 (see also [ChS, Ta]). On the other hand, if we consider more general operators like e.g.

60

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

uniformly elliptic operators on Rd it is only known that Riesz means are uniformly continuous on L1 (X ) if  > d=2 (see [He1]). One of the main points of our work is to investigate the summability of Riesz means for d=2   > (d , 1)=2. The Alexopoulos-Hebisch multiplier theorem discussed above gives optimal value for the exponent d=2 of the number of derivatives needed in spectral multipliers, but it does not give the optimal range of the exponent  for the Riesz summability. Indeed, if k1kWs1 < 1, then   s. However, kR1 kWs2 < 1 if and only if  > s , 1=2. This means that in virtue of Theorem 2 one obtains uniform continuity of Riesz means on Lq for any  > d=2 and for all q 2 (1; 1), whereas Theorem 1 shows Riesz summability for  > (d , 1)=2 (see also [Ch2, pp. 74]). As we mentioned earlier (d , 1)=2 is a critical index for Riesz summability for standard Laplace operator on Rd and Laplace-Beltrami operator on compact manifolds. To conclude we see that the optimal number of derivatives in multiplier theorems is d=2. However, in condition (5) we required d=2 derivatives in L1. In the Hormander-type condition (4) we required d=2 derivatives in L2 . Note that functions t F are compactly supported so condition (5) is strictly stronger than (4). We would like to investigate when it is possible to replace condition (5) in the Alexopoulos-Hebisch multiplier theorem by condition (4) from Theorem 1. As we investigate spectral multipliers in a general setting of abstract operators rather than in a speci c setting of group invariant operators acting on Lie groups, we do not have certain estimates which are consequences of invariant structures of Lie groups. Also we only assume suitable bounds on heat kernels but not pointwise bounds on their space derivatives. The subject of Bochner-Riesz means and spectral multipliers is so broad that it is impossible to provide comprehensive bibliography of it here. Hence we quote only papers directly related to our investigation and refer reader to [Ale1, Ch2, Ch1, ChS, CS, dMM, Du, He1, He3, Ho1, Ho3, HS, MM, SeSo, So1, So2, Si2, St1, St2, Ta] and their references.

2. Main results In this section we rst introduce some notation and describe the hypotheses of our operators and underlying spaces. We then state our main results. Assumption 1. Let X be an open subset of Xe , where Xe is a topological space equipped with a Borel measure  and a distance : Let e (x; y ) < rg be the open ball (of X e ) with centre at B (x; r) = fy 2 X; x and radius r. We suppose throughout that Xe satis es the doubling

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS

61

property, i.e., there exists a constant C such that e 8r > 0: (8) (B (x; 2r))  C(B (x; r)) 8x 2 X; Note that (8) implies that there exist positive constants C and d such that e r > 0: (9) (B (x; r))  C (1 + )d(B (x; r)) 8 > 0; x 2 X; In a sequel we always assume that (9) holds. We state our results in terms of the value d in (9). Of course for any d0  d (9) also holds. However, the smaller d the stronger multiplier theorem we will be able to obtain. Therefore we want to take d as small as possible. Note that in the case of the group of polynomial growth the smallest possible d in (9) is equal to max(d0; d1). Hence our notation is consistent with statements of Theorems 1 and 2. Note that we do not assume that X satis es doubling property. This poses certain diculties which we overcome by using results of singular integral operators of [DM]. An example of such a space X is a domain of Euclidean space Rd. If we do not assume any smoothness on its boundary, then doubling property fails in general. Now we describe the notion of the kernel of the operator. Suppose that T : L1(X; ) ! Lq (X; ) for q > 1. Then by KT (x; y) we denote the kernel of the operator T de ned by the formula

(10) hTf ; f i = 1

2

for all f ; f

2 2

Z

X

Z

Tf f d = 1 2

X

KT (x; y)f (y)f (x) d(x) d(y): 1

2

Cc(X ). Note that T L1 X; !L X; = sup KT ( ; y) L X; : y2X Hence if T L1 X; !L X; < , then its kernel KT is a well de ned measurable function. Vice versa, if supy2X KT ( ; y) L X; < , then KT is a kernel of the bounded operator T : L (X; ) Lq (X; ), even if q = 1. Next we denote the weak type (1; 1) norm of an operator T on a measure space (X; ) by T L1 X; !L1 1 X; = sup  ( x X : Tf (x) >  ), where the supremum is taken over  > 0 and functions f with L (X; ) norm less than one. Assumption 2. Let A be a self-adjoint positive de nite operator. We suppose that the semigroup generated by A on L has kernel 1

k

k

k

k

(

(

)

)

q

(

q

(

)

k

)




k

q

(

)

1

k




k

1

k

j

j

k

(

)

;

(

q

(

1

)

!

f

)

g

1

2

2

62

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

pt(x; y) = K

exp(,tA) (x; y ) which for all t > 0 satis es the following Gaussian upper bound  m=(m,1)  (11) jpt(x; y)j  C(B (y; t)),1=m exp , b (x;t1y=)(m,1) where C; b and m are positive constants and m  2. Such estimates are typical Gaussian estimates for elliptic or subelliptic dierential operators of order m (see e.g. [Da1, Ro, VSC]). We will call pt (x; y) the heat kernel associated with A. When order m = 2, Gaussian estimates (2.4) is equivalent to nite propagation speed, see [Si1]. When m 6= 2, we can have (2.4) but nite propagation speed property does not hold. In our following main results, we suppose that Assumptions 1 and 2 hold. The values d and m always refer to (9) and (11). Theorem 3. Suppose that s > d=2 and assume that for any R > 0 and all Borel functions F such that supp F  [0; R]

Z

jKF mpA (x; y)j d(x)  C(B (y; R, )), kR F kLp X for some p 2 [2; 1]. Then for any Borel bounded function F such that sup ktF kWsp < 1 the operator F (A) is of weak type (1; 1) and is t> bounded on Lq (X ) for all 1 < q < 1. In addition   p kF (A)kL X; !L ;1 X;  Cs sup kt F kWs + jF (0)j : (13) t> Note that if (12) holds for p < 1, then the pointwise spectrum of A is empty. Indeed, for all p < 1 and all y 2 X (12)

(

2

)

1

1

2

0

1(

1

)

(

)

0

(14) 0 = C kf = g kLp = C k a fag kLp  (B (y; 21a )kKfag mpA (
; y)kL2 X; 1 2

p

2

(

)

(

)

so fag ( m A) = 0. Hence for elliptic operators on compact manifolds, (12) cannot be true for any p < 1. To be able to study these operators as well we introduce some variation of assumption (12). Following [CS] for a Borel function F such that supp F  [,1; 2] we de ne the norm kF kN;p by the formula  X  =p N 1 p kF kN;p = N sup jF ()j ; l ,N 2 l,N1 ; Nl where p 2 [1; 1) and N 2 Z . For p = 1 we put kF kN;1 = kF kL1 . It is obvious that kF kN;p increases monotonically in p. The next theorem is a variation of Theorem 3. This variation can be used in the 2

=1

+

1

[

)

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS

63

case of operators with nonempty pointwise spectrum (compare [CS, Theorem 3.6]). Theorem 4. Suppose that  is a xed natural number, s > d=2 and that for any N 2 Z and for all Borel functions F such that supp F  [,1; N + 1] +

Z

(15)

X

jKF

(

mpA) (


; y)j d(x)  C(B (y; 1=N )), kN F kN ;p 2

1

2

for some p  2. In addition we assume that for any " > 0 there exists a constant C" such that for all N 2 Z+ and all Borel functions F such that supp F  [,1; N + 1]

p

kF ( m A)kL1 X; !L1 X;  C"N d "kN F kN ;p: (16) Then for any Borel bounded function F such that supt> kt F kWsp < 1 the operator F (A) is of weak type (1; 1) and is bounded on Lq (X ) for all q 2 (1; 1). In addition 2

(

)

(

+

)

2

1

kF (A)kL

(17)

X;)!L1;1 (X;)

1(



 Cs sup ktF k t>1

Wsp

 1 + kF kL :

Remarks 1. It is straightforward that (12) always holds with p = 1

as a consequence of spectral theory. This means that Alexopoulos' multiplier theorem i.e. Theorem 2 follows from Theorem 3. Theorem 1 also follows from Theorem 3. Indeed, it is easy to check that for homogeneous operators (12) holds for p = 2 (see Section 6 [DOS] or [Ch2, Proposition 3]). 2. The main point of our theorems is that if one can obtain (12) or (15) then one can prove stronger multiplier results. If one shows (12) or (15) for p = 2, then this implies the sharp Hormander-type multiplier result. Actually we believe that to obtain any sharp spectral multiplier theorem one has to investigate conditions of the same type as (12) or (15), i.e. conditions which allow us to estimate the norm kKF mpA (
; y)kL2 X; in terms of some kind of Lp norm of the function F. 3. We call hypotheses (12) or (15) the Plancherel estimates or the Plancherel conditions. In the proof of Theorems 3 and 4 one does not have to assume that p  2 in estimates (12) or (15). However (12) or (15) for p < 2 would imply Riesz summability for  < (d , 1)=2 and we do not expect such a situation. Note that (12) is weaker than (15) and we need additional hypothesis (16) in this case. However, in practice once (15) is proved, (16) is usually easy to check and we can often put " = 0. (

)

(

)

64

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

4. We conclude this paper with a theorem on Riesz summability for

d=2   > (d , 1)=2. Theorem 3 with p = 2 implies Riesz summability for all  > (d , 1)=2 and that in addition it seems that Theorem 3 with p = 2 is essentially stronger than sharp Riesz summability. However, one can obtain only weak type (1; 1) estimates in virtue of Theorem 3

and formally Theorem 3 does not imply continuity and convergence of Riesz means on L (X; ). However, Theorem 3 and 4 can be modi ed to prove that uniform continuity of Riesz means of order greater than (d=2 , 1=p) on all spaces Lq (X; ) for q 2 [1; 1]. We claim the following Theorem. Theorem 5. Suppose that operator A satis es condition (12), or (15) and (16) for some p 2 [2; 1]. Then for any  > d=2 , 1=p and q 2 [1; 1] sup kR (A)kLq X; !Lq X;  C < 1: R> Hence for any q 2 [1; 1) and f 2 Lq (X; ) lim k (A)f , f kLq X; !Lq X; = 0; R!1 R where R is de ned by (7). For the proofs of our Theorems, we refer reader to [DOS]. Here let us only mention that the proofs of Theorems 5 and 3 are less complicated than most of earlier spectral multiplier results. Our strategy is to use the complex time heat kernel bounds (see [Da1, DO]) to show W d = functional calculus for the considered operator A. Then we use Mauceri-Meda interpolation trick (see [MM]) and our Plancherel type assumption (12) to obtain Wd=p " functional calculus. This is enough to show Riesz summability (i.e. Theorem 5. To prove Theorem 3 we need also some Calderon-Zygmund singular integral techniques. However in contrast to the standard Calderon-Zygmund singular integral estimates we do not use estimates for the gradient of the kernel of singular integral operators. Instead of that we follow the ideas of [DM, He3, He2]. 1

(

0

)

(

(

)

)

(

)

2 ( +1) 2

2+

References

[Ale1] G. Alexopoulos. Spectral multipliers on Lie groups of polynomial growth. Proc. Amer. Math. Soc., 120(3):973{979, 1994. [Ale2] G. Alexopoulos. p bounds for spectral multipliers from Gaussian estimates of the heat kernel. preprint, 2000. [ChS] F. M. Christ and C. D. Sogge. The weak type 1 convergence of eigenfunction expansions for pseudodierential operators. Invent. Math., 94(2):421{453, 1988. [Ch1] Michael Christ. Weak type (1 1) bounds for rough operators. Ann. of Math. (2), 128(1):19{42, 1988. L

L

;

SPECTRAL MULTIPLIERS FOR SELF-ADJOINT OPERATORS

65

[Ch2] Michael Christ. Lp bounds for spectral multipliers on nilpotent groups. Trans. Amer. Math. Soc., 328(1):73{81, 1991. [CS] Michael Cowling and Adam Sikora. A spectral multiplier theorem on SU(2). to appear in Math. Z. [CGHM] Michael Cowling, Saverio Giulini, Andrzej Hulanicki, and Giancarlo Mauceri. Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. Studia Mathematica, 111(2):103{121, 1994. [Da1] E. B. Davies. Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989. [Da2] E. B. Davies. Uniformly elliptic operators with measurable coecients. J. Funct. Anal., 132(1):141{169, 1995. [dMM] Leonede De-Michele and Giancarlo Mauceri. H p multipliers on strati ed groups. Ann. Mat. Pura Appl. (4), 148:353{366, 1987. [DO] Xuan Thinh Duong and El-Maati Ouhabaz. Heat kernel bounds and spectral multipliers on spaces of polynomial growth and irregular domains. preprint, 1998. [Du] Xuan Thinh Duong. From the L1 norms of the complex heat kernels to a Hormander multiplier theorem for sub-Laplacians on nilpotent Lie groups. Paci c J. Math., 173(2):413{424, 1996. [DM] Xuan Thinh Duong and Alan McIntosh. Singular integral operators with nonsmooth kernels on irregular domains. Rev. Mat. Iberoamericana, 15(2):233{265, 1999. [DOS] Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora. Plancherel type estimates and sharp spectral multipliers preprint, 2000. [FS] G. B. Folland and Elias M. Stein. Hardy spaces on homogeneous groups. Princeton University Press, Princeton, N.J., 1982. [He1] Waldemar Hebisch. Almost everywhere summability of eigenfunction expansions associated to elliptic operators. Studia Math., 96(3):263{275, 1990. [He2] Waldemar Hebisch. A multiplier theorem for Schrodinger operators. Colloq. Math., 60/61(2):659{664, 1990. [He3] Waldemar Hebisch. Functional calculus for slowly decaying kernels. preprint, 1995. [He4] Waldemar Hebisch. Multiplier theorem on generalised Heisenberg groups. Colloq. Math., 65(2):231{239, 1993. [He5] Waldemar Hebisch. The subalgebra of L1(AN ) generated by the Laplacean. Proc. Amer. Mat. Soc., 117(2):547{549, 1993. [Ho1] Lars Hormander. Estimates for translation invariant operators in Lp spaces. Acta Math., 104:93{140, 1960. [Ho3] Lars Hormander. On the Riesz means of spectral functions and eigenfunction expansions for elliptic dierential operators. In Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965{1966), pages 155{202. Belfer Graduate School of Science, Yeshiva Univ., New York, 1969. [HS] Andrzej Hulanicki and Elias M. Stein. Marcinkiewicz multiplier theorem for strati ed groups. manuscript. [MM] Giancarlo Mauceri and Stefano Meda. Vector-valued multipliers on strati ed groups. Rev. Mat. Iberoamericana, 6(3-4):141{154, 1990. [MS] D. Muller and E. Stein. On spectral multipliers for Heisenberg and related groups J. Math. Pures Appl., 73: 413{440, 1994.

66

XUAN THINH DUONG, EL MAATI OUHABAZ AND ADAM SIKORA

[Ri] Marcel Riesz. Sur la sommation des series de Fourier,. A.S., 1:104{113, 1923. [Ro] Derek W. Robinson. Elliptic operators and Lie groups. The Clarendon Press Oxford University Press, New York, 1991. Oxford Science Publications. [SeSo] A. Seeger and C. D. Sogge. On the boundedness of functions of (pseudo) dierential operators on compact manifolds. Duke Math. J., 59(3):709{736, 1989. [Si1] Adam Sikora. Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2), 47(187):371{382, 1996. [Si2] Adam Sikora. On the L2 ! L1 norms of spectral multipliers of \quasihomogeneous" operators on homogeneous groups. Trans. Amer. Math. Soc., 351(9):3743{3755, 1999. [Si3] Adam Sikora. Multiplicateurs associes aux souslaplaciens sur les groupes homogenes. C.R. Acad. Sci. Paris series 1, 315:417{419, 1992. [SW] Adam Sikora and James Wright. Imaginary powers of Laplace operators. to appear in Proc. Amer. Math. Soc. [So1] Christopher D. Sogge. On the convergence of Riesz means on compact manifolds. Ann. of Math. (2), 126(2):439{447, 1987. [So2] Christopher D. Sogge. Fourier integrals in classical analysis. Cambridge University Press, Cambridge, 1993. [St1] Elias M. Stein. Singular integrals and dierentiability properties of functions. Princeton University Press, Princeton, N.J., 1970. Princeton Mathematical Series, No. 30. [St2] Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [Ta] Terence Tao. Weak-type endpoint bounds for Riesz means. Proc. Amer. Math. Soc., 124(9):2797{2805, 1996. [Tay] Michael Taylor Lp estimates on functions of the Laplace operator. Duke Math. J., 58: 773{793, 1989. [VSC] N. Th. Varopoulos, L. Salo-Coste, and T. Coulhon. Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992. Xuan Thinh Duong, School of Mathematics, Physics, Computing and Electronics. Macquarie University, N.S.W. 2109 Australia

E-mail address :

[email protected]

El Maati Ouhabaz, Institut de Mathematiques Universite de Bordeaux 1 351, Cours de la Liberation 33405 Talence cedex. France

E-mail address :

[email protected]

Adam Sikora, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia

E-mail address :

[email protected]

FROM XY TO ADE DAVID E. EVANS

Abstract. We survey the role of non-commutative operator al-

gebras in statistical mechanics and the relation between the classi cation of modular invariant partition functions in conformal eld theories and braided subfactors.

As exposed in the treatises of Bratteli and Robinson [21] non-commutative operator algebras have a long tradition of providing a framework for understanding quantum statistical mechanics. For example, the one-dimensional XY -model is studied in the Pauli or Fermion algebra Z M2 with the Hamiltonian

N

H=,

Xf(1 + )j j+1 + (1 , )j j+1 + 2j g: j 2Z

x x

y y

z

Here j ;  = x; y; z are the usual Pauli matrices placed at the j th site of the tensor product. Typically one studies time evolution on the Pauli algebra via the one-parameter group of *-automorphisms t = eiHt (
)e,iHt suitably de ned. In such lattice models one is interested in determining the set of equilibrium states, using the Gibbs conditions, KMS condition or a variational principle, minimizing the thermodynamic quantity (energy - temperature.entropy), as well as the return to equilibrium of locally perturbed models. Robinson played a seminal role in this theory, which is described in detail in [21]. Amongst other things, this led to the development of the theory of derivations on operator algebras, the in nitesimal generators of time evolution, which is still relevant today with the Powers-Sakai conjecture [58] a particularly challenging open problem. This led Robinson to working on the in nitesimal generators of (completely) positive semigroups on operator algebras and subsequently his most recent work on heat kernel methods. The Powers-Sakai conjecture asks whether every one-parameter dynamics on a UHF algebra (an in nite tensor product of matrix algebras) or more generally on a simple AF algebra (an inductive limit of nite dimensional algebras) is approximately inner, obtained as a limit of inner 85

86

DAVID E. EVANS

one-parameter groups as in the above XY -example. Kishimoto [49] has recently shown that a stronger form of the conjecture is false, namely the core problem of whether the in nitesimal generator has an AF-core in a suitable sense. If  is a strongly continuous one-parameter group of *-automorphisms of a simple AF algebra, where the domain of the generator is AF, then  is approximately inner. However, [49] constructs on any non type I simple AF C  -algebra examples of approximately inner one-parameter groups of *-automorphisms where the domain of the generator is not AF. These can be regarded as one-parameter continuous analogues of the exotic examples of compact group actions on AF algebras whose xed points are not AF ( rst shown by Blackadar [5] for Z=2 on the Pauli algebra, and latter by Bratteli et al [18] for nite groups and Evans and Kishimoto [33] for compact groups). Returning to our original starting point of this paper, the XY -model, notice that it degenerates at certain values of (; ), namely at (0; 1), to the Ising nearest neighbour model. This is a classical Hamiltonian, and it would therefore appear to be arti cial to study it via a noncommutative framework, the Pauli algebra. Nevertheless, there is a natural role for non-commutative operator algebras in the study of such classical statistical mechanical models which is the point of this present survey. This connection begins with the transfer matrix method. Let us take a two dimensional nearest neighbour Ising model on a square lattice Z2 with Hamiltonian H =,

X J

; nn

with the summation over the vertices or sites ; on Z2 which are nearest neighbours (nn). We switch from one to two dimensions because the one dimensional version does not have a phase transition at a non zero temperature. At each site  or vertex point of the lattice we have a variable, a spin or magnetization  with either a positive or negative orientation or value represented by +1 or ,1. Then a state  = ( ) of the Ising model is a distribution of pluses and minuses over the square lattice L = Z2, so any con guration is represented by a point in con guration space the compact Hausdor space P = f1gL. Thus the natural home to study this Ising model is the space C (P ) = Z2 C 2

N

FROM XY TO ADE

87

the commutative C -algebra of all continuous functions on the compact con guration space. At each inverse temperature we may be interested in the simplex K of equilibrium states, given say by solutions to the equations of Dobrushin, Lanford and Ruelle [30][50] or the variational principle: minimize(energy - temperature.entropy). In the algebraic approach, one uses the transfer matrix formalism to transform the setting to that of a one dimensional quantum model, represented by a non commutative "one dimensional" C  -algebra and time evolution t . The transfer matrix T is obtained for the partition function of a strip of nite length M and width length one. With boundary conditions ;  along the two lengths the corresponding partition function T de nes us the transfer matrix T . The partition function Z of a nite rectangular lattice of length M and width N is then obtained by multiplying the strip partition functions, namely transfer matrix entries and summing over internal edges. For periodic boundary conditions we obtain (1)

Z=

X exp(, H ()) = X T

1 2 T 2 3

: : : TN 1 = trace T N :

N

N

In this way we move from the commutative algebra C (P ) = Z2 C 2 to the non-commutative Pauli algebra A = Z M2 where the local transfer matrices T generate the even part A+. Time evolution can be formally written as t = T it(
)T ,it, i.e. we consider T = e,H where H is now a quantum Hamiltonian which is no longer a (one dimensional) Ising Hamiltonian. Spatial translation by Z2 in the classical model Z2 C 2 corresponds to spatial translation in AP = Z M2 together with an evolution fT n(
)T ,n : n 2 Zg in the orthogonal transfer direction. For each inverse temperature one looks for a map F ! F from (local) classical observables in C (P ) to the quantum algebra A, and a map  ! ' from states on C (P ) (or measures on P ) to linear functionals on the local observables in AP such that one can recover the classical expectation values or correlation functions from a knowledge of the quantum ones alone: (F ) = ' (F ). Fixing some boundary conditions, then for each inverse temperature , let ' denote the corresponding state on A. [In general positivity of ' is not automatic but follows from re ection positivity of ]. Then if c denotes the inverse critical temperature of Onsager, there exist automorphisms

N

N

88

DAVID E. EVANS

f : 6= cg of A [34] which do not depend on boundary conditions, and real analytic in = 6 c such that (

' = '1   > c '0   < c: N

p

,


Here '0 = Z ! , where = 11 N2 is the disordered state, and ,for
,1
, +  + or - boundary conditions, '1 = Z !  where = 0 ; = 01 respectively and '1 = ('+1 + ',1) =2 for free or periodic boundary conditions. Thus with free or periodic boundary conditions, we conclude that ' is pure for 0  < c (also for = c by dierent methods [2]) and is a mixture of two inequivalent pure states for ' for > c. If h i denote the classical states corresponding to + and - boundary conditions respectively, then we can deduce that hF i is real analytic in > c when F is a local classical observable, as hF i = '1 (F ) using analyticity of  . A dynamical system t on AP is formally given as t = T it(
)T ,it which has a unique ground state for < c and two extremal ground states ' for > c. The Ising model can be generalized to the possibility of having more than two values or spins possible at any lattice site, and moreover some constraints or rules to determine allowable con gurations. A particular value at one site may force only restricted choices at nearest neighbours. This would be achieved by distributing values of a xed graph G at sites of the lattice L in such a way that if  and are nearest neighbours in L, then the corresponding values  and  are joined in the graph. The state space P can be de ned for any graph, but if G contains some multiple edges, we consider distributions of edges of G on edges of L. For the Dynkin diagram A3 with vertices labelled by f; g and square lattice L one obtains two copies of the Ising model as in Fig. 1 by placing the frozen spin  on the even or odd sublattices of L. This graph may be generalised to the Dynkin diagrams of Fig. 1 for the models of Andrews, Baxter and Forrester [1]. These in turn can be generalised by considering the Weyl alcove A(n;k) of the level k integrable representations of the Kac-Moody algebra SU (n)^ : Boltzmann weights associated to a local con guration around minimal squares of the lattice can be chosen to satisfy the integrable YangBaxter equation [25].

FROM XY TO ADE

 r

,,A@+ r

3

 r



 r

 r

 r

 r



 r

 r

89

 r

 r



Ising model: Dynkin diagram A and con guration space Figure 1.

3

The centre of SU (n), the abelian Z=n, acts on A n;k , e.g. Z=2 on the Dynkin diagram Ak by a ip i ! k , i which may or may not have a xed point depending on the parity of k. The interesting case is when there is a xed point. In any case, the Boltzmann weights are preserved under the symmetry, and yield new Boltzmann weights on the orbifold graphs A n;k =(Z=p), whenever p divides n, satisfying the integrable Yang Baxter equation [28] [35]. For example, when n = 2; k = 2m we blow up the xed point m to a pair (a copy of Z=2) and replace each distinct pair i; k , i (i 6= m) interchanged by the symmetry with a singleton yielding the graph Dm . The case A is self dual in that A =(Z=2) = A . Nevertheless, the situation here is not entirely trivial. This is Kramers-Wannier high temperature{low temperature duality. This duality replaces the Boltzmann weights at a temperature t with ones at dual temperature t. Again the xed point of the symmetry t ! t is what provides the interesting structure | at the critical temperature tc of Onsager. We have mentioned the phenomena of AF algebras with non-AF xed point algebras under symmetries. Such examples were rst found using similar orbifold constructions. As a continuous version of the ip on a Dynkin diagram which yields symmetries on AF algebras, consider instead the ip on the interval around its midpoint or a ip on a circle around an axis in its plane through its centre. The orbifold space is best described by taking the cross product. For a pair of points interchanged by the symmetry, the local crossed product is simply a two by two matrix algebra. The diagonal elements represent the continuous functions on the pair, and the o-diagonal elements come from the transition between the two points. Each xed point is replaced by a pair arising from the transitions only generating a copy of C as the continuous functions on the group (dual). Thus gluing together, we (

)

+1

(

)

+2

3

3

3

2

90

DAVID E. EVANS

represent the action of the ip on the interval [0; 1] by the C -algebra of continuous functions on a half-interval [0; 1=2], which on the half-open half-interval [0; 1=2) (the part of the interval which has a non-trivial orbit) are two by two matrix valued but at other end point 1=2 are diagonal. So the spectrum of this algebra is a continuous version of the D-Dynkin diagram. Topologically it is an interval with two non Hausdor points at one end. The analogous action of the ip on the two torus which conjugates each variable has four xed points, yielding a sphere with four singular points. The corresponding cross product is the space of two by two matrix valued functions on the sphere which are diagonal at four distinguished points. Replacing the two torus by a non-commutative torus generated by two unitaries U and V satisfying the commutation relation V U = qU V where q = exp2i we obtain a non commutative toroidal orbifold when taking the symmetry which inverts the generators U and V . It is Morita equivalent to the algebra of a singular ow on a sphere obtained as the quotient of the Kronecker ow on the torus as illustrated in [36], page 137 or http://www.cf.ac.uk/maths/opalg/ncto/. Remarkably these algebras are AF (when  is irrational)([20][19] or [32],[65]) although the corresponding irrational rotation algebras and algebras of the Kronecker ow are not. The non-commutative torus has a representation on L2 (T2 ) where U and V are represented as multiplication and translation operators. In this representation, or at least if one takes the Fourier transform, the Hamiltonian H = U + U ,1 + (V + V ,1 ) are the Mathieu or discrete Schrodinger operators with almost periodic potentials. The natural home to study these operators is the xed point algebra under our ip because when  is irrational then U + U ,1 and V + V ,1 generate the xed point algebra. It is still a tantalizing mystery as to whether there is a relation between the AF property of the xed point algebras, a strong form of non-commutative disconnectedness, and the Cantor spectra of such almost Mathieu operators - which are at least known to be Cantor for generic coupling constant  and rotation number . Symmetries on such algebras, where there are underlying xed points can produce algebras with totally dierent properties. Similarly, symmetries on subfactors, statistical mechanical models, conformal eld theories can produce totally dierent subfactors etc from what one started with.

FROM XY TO ADE

91

From a lattice model one may obtain a eld theory by taking a continuum or scaling limit, letting the lattice spacing go to zero whilst simultaneously approaching the critical temperature. As the scale or correlation length becomes in nite, one obtains a scale invariant or conformal invariant theory. Belavin et al [4] suggested that the scale invariance at a critical point is enhanced to conformal invariance. One of the invariants of the conformal eld theory is the central charge a multiplier in projective representations Lm of the vector elds ,zm+1 d=dz on the circle, the Virasoro algebra. However the central charge can already raise its head in the statistical mechanical model. Going back to the partition function Z of Eq. (1) the free energy f = ,logZ=NM is independent of boundary conditions as N; M ! 1. However the asymptotics depend on boundary conditions; if 1

93

0 and T is diagonal:  (

X

,1= ) =



S  ( );  (

+ 1) =

X 

T  ( ):

Then the classi cation of modular invariant partition functions can be reformulated as a matrix problem. Find all matrices Z subject to Eq. (2) commuting with S and T . This is a rather concrete problem. For SU (2) at level k, SU (2)k , the admissible weights are the spins  = 0; 1; ::; k and the Ka c-Peterson matrices are given explicitly as r 2  ( + 1)( + 1) S = sin k+2 k+2



( + 1) ,  i T =  exp i 2k + 4 4 with ;  = 0; 1; :::; k, and the characters as  (q )

=

X

2 q (+1) =4(k+2)

 (q )

3

2



(2n(k + 2) +  + 1)qn(n(k + 2) +  + 1)

n2Z

Q

n if q = e i , and the Dedekind function (q) = q = 1 n (1 , q ): For the Ising model, the characters are (in the notation of Fig. 1), 2

1 24



= [# =2] = ;  = ([# =] =  [# =n] = )=2 3 2

2

p

#3

p

#4

= = q ,1=48

= = q ,1=48

p

#2 =

3 2

3

with the #-functions:

=

q

1 , Y n=0

1 , Y n=0

1=24

1 Y

3 2

4

1 + qn

+1=2

1 , qn

+1=2





(1 + qn):

n=1

Here the Kac-Petersen matrices are simply 0 1 p2 1 1 p p S= @ 2 0p , 2 A ; T = ei= 1 , 2 1 1 2

=1

24

01 @0 0

0

ei3=8

0

0 0

ei

1 A

94

DAVID E. EVANS

so that there is only one modular invariant the diagonal mass matrix or: ,
Z = j#2j3 + j#3 j3 + j#4 j3 =2jj3: Whilst the mass matrix is trivial, the partition function itself has some structure. The following is also a modular invariant particular function Z = (j#2 j + j#3j + j#4 j)=2jj L this time for the coset model su(2)1 su(2)1=su(2)2, which also exhibits Ising fusion rules (as does (E8)1  (E8 )1=(E8 )2) and so(5)1). At rst sight, it might appear that generally there may be an in nite number of solutions to this modular invariant problem. However, there is a following estimate [12]: P Z  dd which is a strengthening of the inequality of Gannon [42]: Z  1=S002 if d = S0=S00 : Thus since Z is positive and integral there are at most nitely many solutions, for a xed representation of SL(2; Z). In the case of SU (2), there are at most three solutions for a xed level k. This is the ADE classi cation of Capelli, Itzykson and Zuber [22]. A Dynkin diagram is associated to each invariant through the identi cation of diagonal terms of the invariant f : Z 6= 0g = I with eigenvalues fSf=S00 :  2 Ig of the corresponding Dynkin diagram if f = 1 the fundamental representation of SU (2). The A refers to the diagonal invariant, D to orbifold invariants and E to the three E6; E7 ; E8 exceptional invariants. For SU (3) there is an anologous ADE classi cation due to Gannon [43]; di Francesco and Zuber [28] sought to show systematically the existence of graphs with spectra matching the modular invariant, give a meaning to these graphs themselves and compute them in a number of examples. As we have said there are at most nitely many solutions to the modular invariant conditions. There is always one solution the trivial diagonal invariant: X 2A

jj2

where the corresponding mass matrix is diagonal Z =  . In some sense, [52] [29] [11] 'every' modular invariant is diagonal if looked at properly. If we can extend the A system to a B system so that the characters decompose X  = b  ;  2 B 2A

FROM XY TO ADE

95

according to some branching rules, then the diagonal B-modular invariant will give an A-modular invariant j j = j b j :

X

XX

2

 2B

2

 2B 2A

In some sense, every modular invariant should look like this or with a possible twist   w  ; for a permutation w of the extended fusion rules, preserving the vacuum. The problem in general is then to nd such extensions. When there is no twist present we have what are sometimes called type I invariants: Z = b; b : ( )

X 

These are automatically symmetric: Z = Z . In the presence of a non-trivial twist, we have the type II invariants Z = b bw   :

X

( )



These are not necessarily symmetric, but at least there is symmetric vacuum coupling Z  = Z . Not every modular invariant is even symmetric in this sense, (e.g. for SO(16n) ) but every known SU (n) modular invariant is even symmetric in the usual sense. Our aim is to study or even construct modular invariants from subfactors. The framework can be summarised as follows. We have a hyper nite III factor N on which there is a system of endomorphisms f 2 Ag labelled by our positive energy representations or our original states in the original statistical mechanical setting. We induce these endomorphisms to endomorphisms  on a larger ambient factor M | there will be two natural ways to do this labelled . The modular invariant will then be constructed or recovered as Z = h ; , i where the right hand side will be computed as dimensions of intertwiner spaces or the number of common sectors when we decompose  into irreducibles. The original endomorphism  2 A will be irreducible but  may not be. The factor N will carry the modular data for S and T matrices, varying the inclusion may change the modular invariant but somehow the inclusion will have to be related to the original A-system. The system A on the factor N can be constructed via the method of Jones-Wassermann. First for any positive energy representation 0

0

1

1

+

96 

DAVID E. EVANS

2 A, the objects LI SU (n) and LI SU (n), if LI ; LI denote loops 0

0

on SU (n) concentrated on complementary non-trivial intervals I and I in the circle. We can thus form the inclusion (3) LI SU (n)  LI SU (n) : For  = 0, the vacuum representation, there is Haag duality and this inclusion is not proper but gives us a single hyper nite III1 factor N (more precisely a net N (I ) of such factors). The inclusion Eq. (3) then determines a system of endomorphisms  2 A, so that the inclusion Eq. (3) is isomorphic to N  N , with index [N , N ] = d2. Wassermann [66] has shown that the fusion rules of such endomorphisms are precisely
, the same as that of SU (n) at a root of unity q = e2i=(k+n) . Moreover, rotation through 1800 on the circle, interchanges the role of I and I . This has the eect that the system A is naturally braided, i.e. not only is the system commutative  =  as sectors if ;  2 A  End(N ), but there is a choice "(; ) of unitaries taking  to  satisfying the Yang Baxter equation, braiding fusion equation etc.  ] ;  2 A, deterThus we have commutative matrices N = [N ; mining the fusion X    = N 0

00

0

0

0



with composition of endomorphisms, or rather sectors, their unitary equivalence classes and a natural notion of addition. Fusion by the endomorphism of the conjugate  of  is given by N = Ntr, the transpose. Thus fN :  2 Ag is a family of commuting normal matrices and so simultaneously diagonalisable. By the Verlinde theory the unitary matrix which performs this diagonalisation is the S matrix itself. Inverting the consistency condition or the regular representation X  N (4) N N = N  we obtain X S  = N S S S  

or



N =

0

X S 

S0

jSihSj:

The modular invariants will provide representations other than the regular representation Eq. (4), and pick out subsets fS=S0 :  2 Ig

FROM XY TO ADE

97

where I = f : Z 6= 0g; the diagonal part of the modular invariant. At the same time these representations will replace N by other families G of graphs | to be associated with the modular invariant or at least the subfactor N  M which yields that particular invariant. As we have said the inclusion, which is meant to duplicate the modular invariant, should be related to the original A-system. This is achieved as follows. There is a conjugation on endomorphisms of N , (extending for groups the notion of inverting automorphisms or conjugating a representation in the dual) compatible with the conjugation on A. Similarly one can conjugate endomorphisms or sectors of M , or those between N and M , M and N . In particular, we can take the inclusion  = N ! M , its conjugate  = M ! N and compose to get endomorphisms =  on M and  = jN =  on N called the canonical and dual canonical endomorphisms respectively. What we need is  lies in the system generated by A, i.e. decomposes as a sum of sectors from A. Note that we do not need to specify M when we ask whether a particular endomorphism  of N is a dual canonical endomorphism. It may not be particularly clear in a given situation whether a certain endomorphism is a dual canonical endomorphism or what M may be. When Z is a modular invariant typical candidates for dual canonical endomorphisms will be  Z0 ;  Z0  on N and Z  opp on N N opp where N opp is the opposite algebra, etc. The rst non trivial (i.e. exceptional) invariant for SU (2) occurs at level 10:

P

N

(5)

ZE6

P

P

N

= j0 + 6j2 + j4 + 10 j2 + j3 + 7 j2:

The diagonal part of the invariant I = f : Z 6= 0g matches the spectrum of the Dynkin diagram E6 , namely fS1=S0 = 2 cos ( +1)=12 :  = 0; 6; 4; 10; 3; 7g: For this reason Capelli, Itzykson and Zuber labelled the invariant by the graph E6 . In the subfactor setting we can derive this graph as follows. First, we turn to the conformal embedding description of this invariant due to Bouwknegt and Nahm [17] which provides the extended system B which diagonalises the invariant. The embedding SU (2)10  SO(5)1 means there is a mapping of SU (2) in SO(5) such that the three level 1 representations B of SO(5) decompose into level 10 representations of SU (2) with nite multiplicity. The

98

DAVID E. EVANS

system SO(5)1 has three inequivalent representations, b,v,s basic, vector and spinor which reproduce the Ising fusion rules. They decompose (cf Eq. (5)) as (6) b = 0 + 6 ; v = 4 + 10 ; s = 3 + 7 so that the E6 modular invariant for SU (2)10 arises from the diagonal invariant for SO(5)1: ZE6 = jbj2 + jv j2 + jsj2: Moving now to the loop group factors the conformal embedding gives us an inclusion of factors: L SU (2)  LSO(5) using the vacuum representation on LSO(5), a net of subfactors N (I )  M (I ): Fixing I, we have subfactor N  M on which there are systems A = SU (2)10 and B = SO(5)1 of endomorphisms acting respectively. These two systems can be related by a form of Mackey inductionrestriction which in the subfactor setting goes back to Longo-Rehren [51]. Using the braiding "+ or its opposite braiding ",, we can lift endomorphisms  in A to those of M:  = ,1Ad"(; ) . The maps [] ! [] preserve all the operations of conjugation, addition and multiplication of sectors [67][8][9][10]. However, they are not injective, and  may be reducible. We nd that f+ :  2 Ag decomposes into six irreducible sectors such that the graph E6 is multiplication by 1+ [67] [9]. In fact [1+] = G; is part of a system of matrices with non-negative entries fG :  2 Ag which represents the original A-fusion rules. This had been noticed empirically in e.g. [28] [55] which now gets a subfactor explanation. To bring the B system into the game we use -restriction,  =   to take M -sectors to N -sectors. This map  is not multiplicative, but in the type I situation there is a reciprocity: h; i = h;  i (with inequality on the type II setting) as long as say is a subsector of the induced system E6 respectively. Since  restriction takes the E6 systems into the A-system by Eq. (6), namely b =  = 0 + 6 ; v = 4 + 10 ; s = 3 + 7 the reciprocity means that the B system, coming from the three level 1 representations of SO(5) must lie in the induced systems E6, i.e.

FROM XY TO ADE

99

A3 = B  E6+ \ E6,. In fact we can identify as sectors b = 0; v = 10; s = 3 , 9; and E6+ \ E6, = A3 precisely. The dual canonical endomorphism  lies in A, but a priori we do not have much information about its Fourier transform . In fact as sectors: = idM + 1+1, so that 2 E6+ _ E6,, the full induced system. Indeed A = E6+ _ E6, , the system generated by E6 is precisely all subsectors of f :  2 Ag in fact the latter has global dimension w = A d2, whilst if w = 2 2A d denotes the global dimension of the induced system then [10]:

P

w=w =

P

XZ d

0 

with the sum over only the degenerate sectors in A | which have trivial monodromy with all other sectors. In this case the A-system, as far SU (n)k is non-degenerate, the vacuum is the only degenerate sector. Moreover we can recover the modular invariant as

Z = h+; ,i; ;  2 A: In this case the E6 systems are commutative (but not braided) as is the E6+ _ E6, system | but this is not always the case. The neutral system A0 = A+ \ A,, if A are the induced chiral systems, is braided, with the braiding non-degenerate if that of A is. Complexifying the nite dimensional algebras A we can decompose them in the non-degenerate case as [15]: (7)

A =

M M Mat(b ): 



 2A0 2A

Here b; are the chiral branching coecients h; i;  2 A;  2 A. (In the case of chiral locality where the extended net M (I ), is local, i.e. observables associated with disjoint intervals commute, then b; = h; i = h ; i;  2 A0;  2 A:) In particular the extended systems are commutative only when b;  1;  2 A0;  2 A Thus the informal inclusions SU (n)n  SU (n2 ,1)1 give non-commutative chiral systems when n  4; and it explains the computations of Feng Xu [67] who found non-commutativity in case n = 4 by a direct computation. Thus we can decompose the modular invariant as

Z = h+; ,i =

Xb

 2A0

 b

; ;

100

DAVID E. EVANS

and from Eq. (7) by counting dimension we see that

jAj =

X(b ) ;

= trbt b :

2

In the case of chiral locality where b = b, , so that the invariant is type I, we see that jAj = trZ t Z , more generally jAj only sees the type I parent of a type II invariant. Thus with +

ZD10 = j0 + 16 j2 + j4 + 12 j2 + j6 + 10 j2 + 2j8 j2 + j2 + 14 j2 ZE7 = j0 +16 j2+j4 +12 j2+j6 +10 j2+j2 j2+(2 +14 ),8 +8 (2 +14 ),

then in either case trb b = 10 so that multiplication by [] gives the graph D in either case so we do not get the graph E for ZE7 (where we can use the dual canonical endomorphisms  +  ,  +  +  respectively). In general (and this will work when either chiral locality holds or fails) we look at the action of A on the M -N sectors M AN which are the irreducible sectors of  =  (which can be identi ed with A when chiral locality holds). This action decomposes as:  Mat(Z ), with S  = 1 : S Z 1

10

7

0

16

0

16

8

L

M 



0

Thus we get the desired representation with spectrum matching the diagonal part of the modular invariant, and counting dimension then jM AN j = trZ; e.g. trZE7 = 7 so that we do indeed now recover the correct graph. The subfactor framework is rich enough to produce a Moore-Seiberg type decomposition of modular invariants as well as handle possibly non-symmetric modular invariants. As we have already observed, in the case of chiral locality, b; = b,;(= h;  i) for  2 A;  2 A : So the question arises as to how far we can identify b and b, , say b,; = bw  ; for a permutation w of the extended neutral system B or if we need dierent labellings B or B, to handle possibly non-symmetric modular invariants. Now locality holds if and only if  = Z  = Z  : In general we de ne  = Z ; , = Z  : Using the theory of intermediate subfactors of [45], we can show [16] that both  are dual canonical endomorphisms for inclusions N  M which satisfy chiral locality and M  M . This means we can use -induction on both +

0

+ ( )

+

+

+

P

0

P

P

0

0

P

0

FROM XY TO ADE

101

inclusions N  M, to obtain type I modular invariants Z , such that Z = Z  = Z ; Z, = Z , = Z ; where we can identify both neutral systems M A M with M AM . If M = M, ; then we can write  = b b Z + 0

+ 0

0

0

0

0

0

0

X

+

 2A0

in particular (Z = Z ,) and using the identi cations M AM with M AM to produce an automorphism ! of the neutral elements M AM we have: Z = b bw   : In the case of E invariant we have N  M  M where M = M, and the dual canonical endomorphism for N  M ; N  M are  +  ,  +  +  as we have said before. It may happen that M 6= M, and this does occur for SO(16n) where there are non-symmetric modular invariants where we must use dierent labelling M+ AM+ ;M, AM, on the left and right to decompose , Zext + ;, as , ;# + , where # = #, # is the identi cation. The situation is summarised [10] using recent work of Rehren [62] on canonical tensor product subfactors as a pair of inclusions: +

0

0

0

X

( )

7

0

16

0

+

16

8

+

1

0

(

N

0

1

)

+

N

ON

P

opp

M

N

+

OM P

opp

,

B

N N

where the dual canonical endomorphisms for N N opp  B and M M,opp  B as Z  opp;  2A0 # ( ) #, ( )opp respectively. There is a connection between the two chiral inductions and the picture of left- and right-chiral algebras in conformal eld theory. Suppose that our factor N is obtained as a local factor N = N (I ) of a quantum eld theoretical net of factors fN (I )g indexed by proper intervals I  R on the real line, and that the system N XN is obtained as restrictions of DHR-morphisms (cf. [44]) to N . This is in fact the case in our examples arising from conformal eld theory where the net is de ned in terms of local loop groups in the vacuum representation. Taking two copies of such a net and placing the real axes on the light cone, then this de nes a local net fA(O)g, indexed by double cones O on two-dimensional Minkowski space (cf. [61] for such constructions). Given a subfactor N  M , determining in turn two subfactors N  M obeying chiral +

+

102

DAVID E. EVANS

locality, will provide two local nets of subfactors fN (I )  M(I )g as a local subfactor basically encodes the entire information about the net of subfactors [51]. Arranging M+ (I ) and M,(J ) on the two light cone axes de nes a local net of subfactors fA(O)  Aext (O)g in Minkowski space. The embedding M+ M,opp  B gives rise to another net of subfactors fAext (O)  B (O)g, where the net fB (O)g obeys local commutation relations. The existence of the local net was already proven in [62], and now the decomposition of [ext ] tells us that the chiral extensions N (I )  M+(I ) and N (I )  M, (I ) for left and right chiral nets are indeed maximal (in the sense of [61]), following from the fact that the coupling matrix for fAext (O)  B (O)g is a bijection. This shows that the inclusions N  M should in fact be regarded as the subfactor version of left- and right maximal extensions of the chiral algebra.

References [1] Andrews, G. E., Baxter, R. J., and Forrester, P. J.: Eight vertex SOS model and generalized Rogers{Ramanujan type identities. J. Stat. Phys., 35, 193{266 (1984). [2] Araki, H., Evans, D.E.: On a C  -algebra approach to phase transition in the two dimensional Ising model. Commun. Math. Phys. 91, 489-503 (1983). [3] Behrend, R.E., Pearce, P.A., Petkova, V.B., Zuber, J.-B.: Boundary conditions in rational conformal eld theories Nucl. Phys. B570, 525-589 (2000). [4] Belavin, A. A., Polyakov, A. M., and Zamolodchikov, A. B. In nite conformal symmetry in two-dimensional quantum eld theory. Nucl. Phys. B, 241, 333{ 380 (1980). [5] Blackadar, B.: Symmetries of the CAR algebra. Annals of Math., 131, 589{623 (1990). [6] Bockenhauer, J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265-304 (1996). [7] Bockenhauer, J.: An algebraic formulation of level one Wess-Zumino-Witten models. Rev. Math. Phys. 8, 925-947 (1996). [8] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. I. Commun. Math. Phys. 197, 361-386 (1998). [9] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. II. Commun. Math. Phys. 200, 57-103 (1999). [10] Bockenhauer, J., Evans, D.E.: Modular invariants, graphs and -induction for nets of subfactors. III. Commun. Math. Phys. 205, 183-228 (1999). [11] Bockenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267-289 (2000).

FROM XY TO ADE

103

[12] Bockenhauer, J., Evans, D. E.: Subfactors from modular invariants. math.OA/0006114. [13] Bockenhauer, J., Evans, D. E.: Subfactors and modular invariants. math. OA/0008056. [14] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: On -induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429-487 (1999). [15] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733-784 (2000). [16] Bockenhauer, J., Evans, D.E., Kawahigashi, Y.: Longo-Rehren subfactors arising from -induction. Publ. RIMS, Kyoto Univ. to appear, math.OA/0002154. [17] Bouwknegt, P., Nahm,W.: Realizations of the exceptional modular invariant A(1) partition functions. Phys. Lett. B, 184, 359-362 (1987). 1 [18] Bratteli, O., Elliott, G. A., Evans, D. E., and Kishimoto, A.: Finite group actions on AF algebras obtained by folding the interval. K -theory, 8, 443{464 (1994). [19] Bratteli, O., Evans, D. E. and Kishimoto, A.: Crossed products of totally disconnected spaces by Z2  Z2. Ergodic Theory and Dynamical Systems, 13, 445{484 (1994). [20] Bratteli, O. and Kishimoto, A.: Non-commutative spheres, III, Irrational rotations. Commun. Math. Phys., 147, 605{624 (1992). [21] Bratteli,O., Robinson,D.W.: Operator algebras and quantum statistical mechanics. I and II 2nd editions Springer, Berlin (1987 and 1997) [22] Cappelli, A., Itzykson, C., Zuber, J.-B.: The A-D-E classi cation of minimal and A(1) 1 conformal invariant theories. Commun. Math. Phys. 113, 1-26 (1987). [23] Cardy, J. L., Operator content of two-dimensional conformally invariant theories. Nucl. Phys. B270 186-204 (1986). [24] Carey, A. L., Evans, D. E.: The operator algebras of the two dimensional Ising model. In: Birman, J. et al (eds.): Braids. Contemp. Math. 78, 117-165 (1988). [25] Date, E., Jimbo, M., Miwa, T., Okado, M.: Solvable lattice models. In: Theta functions { Bowdoin 1987, Part 1. Proc. Symp. Pure Math. 49, Providence, R.I.: American Math. Soc., pp. 295-332 (1987) [26] Davies, E.B., Pearce, P.A.: Conformal invariance and critical spectrum of corner transfer matrices. J. Phys. A23, 1295-1312 (1990). [27] Di Francesco, P., Mathieu, P., Senechal, D.: Conformal eld theory. New York: Springer-Verlag 1996. [28] Di Francesco, P., Zuber, J.-B.: SU (N ) lattice integrable models associated with graphs. Nucl. Phys. B338, 602-646 (1990). [29] Dijkgraaf, R., Verlinde, E.: Modular invariance and the fusion algebras. Nucl. Phys. (Proc. Suppl.) 5B, 87-97 (1988). [30] Dobrushin,R. L.: Gibbsian random elds for lattice systems with pair-wise interactions. Functional Analysis and its applications, 2, 292-301 (1968).

104

DAVID E. EVANS

[31] Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations. II. Commun. Math. Phys. 15, 173-200 (1969). [32] Elliott, G. A. and Evans, D. E.: The structure of the irrational rotation C  algebra. Annals of Mathematics, 138, 477{501 (1993). [33] Evans, D. E. and Kishimoto, A.: Compact group actions on UHF algebras obtained by folding the interval. J. Funct. Anal., 98, 346{360 (1991). [34] Evans, D. E., Lewis, J.T.: On a C  -algebra approach to phase transition in the two dimensional Ising model. II. Commun. Math. Phys. 102, 521-535 (1986). [35] Evans, D. E., Kawahigashi, Y.: Orbifold subfactors from Hecke algebras. Commun. Math. Phys. 165, 445-484 (1994). [36] Evans, D. E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press 1998. [37] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. Commun. Math. Phys. 125, 201-226 (1989). [38] Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. Rev. Math. Phys. Special issue, 113-157 (1992). [39] Frohlich, J., Gabbiani, F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251-353 (1990). [40] Frohlich, J., Gabbiani, F.: Operator algebras and conformal eld theory. Commun. Math. Phys. 155, 569-640 (1993). [41] Fuchs, J., Schellekens, A.N., Schweigert, C.: Galois modular invariants of WZW models. Nucl. Phys. B437, 667-694 (1995). [42] Gannon, T.: WZW commutants, lattices and level{one partition functions. Nucl. Phys. B396, 708-736 (1993). [43] Gannon, T.: The classi cation of ane SU (3) modular invariants. Commun. Math. Phys. 161, 233-264 (1994). [44] Haag, R.: Local Quantum Physics. Berlin: Springer-Verlag 1992. [45] Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155, 25-63 (1998). [46] Jimbo, M.: A q-analogue of U (N + 1), Hecke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11, 247-252 (1986). [47] Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1-25 (1983). [48] Kawahigashi, Y.: On atness of Ocneanu's connections on the Dynkin diagrams and classi cation of subfactors. J. Funct. Anal. 127, 63-107 (1995). [49] Kishimoto, A.: Examples of one-parameter automorphism groups of UHF algebras. math.OA/0008003. [50] Lanford, O.E., Ruelle, D.: Observables at in nity and states with short range correlations in statistical mechanics. Commun. Math. Phys., 13, 194215 (1969).

FROM XY TO ADE

105

[51] Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567-597 (1995). [52] Moore, G., Seiberg, N.: Naturality in conformal eld theory. Nucl. Phys. B313, 16-40 (1989). [53] Nahm,W.: Lie group exponents and SU (2) current algebras. Commun. Math. Phys.,118, 171-176,(1988). [54] Ocneanu, A.: Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors. (Notes recorded by S. Goto) In: Rajarama Bhat, B.V. et al. (eds.), Lectures on operator theory, The Fields Institute Monographs, Providence, Rhode Island: AMS publications 2000, pp. 243-323. [55] Petkova, V.B., Zuber, J.-B: From CFT to graphs. Nucl. Phys. B463, 161-193 (1996). [56] Petkova, V.B., Zuber, J.-B.: Conformal eld theory and graphs. In: Proceedings Goslar 1996 \Group 21", hep-th/9701103. [57] Popa, S.: Classi cation of subfactors of nite depth of the hyper nite type III1 factor. Comptes Rendus de l'Academie des Sciences, Serie I, Math., 318, 10031008 (1994). [58] Powers, R.T., Sakai, S.: Existence of ground states and KMS states for approximately inner dynamics. Commun. Math. Phys., 39, 273-288 (1975). [59] Rehren, K.-H.: Braid group statistics and their superselection rules. In: Kastler, D. (ed.): The algebraic theory of superselection sectors. Palermo 1989, Singapore: World Scienti c 1990, pp. 333-355. [60] Rehren, K.-H.: Space-time elds and exchange elds. Commun. Math. Phys. 132, 461-483 (1990). [61] Rehren, K.-H.: Chiral observables and modular invariants. Commun. Math. Phys. 208, 689-712 (2000). [62] Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395-406 (2000). [63] Rehren, K.-H., Stanev, Y.S., Todorov, I.T.: Characterizing invariants for local extensions of current algebras. Commun. Math. Phys. 174, 605-633 (1996). [64] Verstegen, D.: New exceptional modular invariant partition functions for simple Kac-Moody algebras. Nucl. Phys. B346, 349-386 (1990). [65] Walters, S. G.: Inductive limit automorphisms of the irrational rotation C  algebra. Commun. Math. Phys., 171, 365{381 (1995). [66] Wassermann, A.: Operator algebras and conformal eld theory III: Fusion of positive energy representations of LSU (N ) using bounded operators. Invent. Math. 133, 467-538 (1998). [67] Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 347-403 (1998). School of Mathematics, University of Wales Cardiff, PO Box 926, Senghennydd Road, Cardiff CF24 4YH, Wales, U.K., EvansDE@cardi.ac.uk

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY ROBERT GULLIVER

Contact geometry treats such questions as the existence and classi cation of contact structures on manifolds of odd dimension and speci ed topological structure. See inequality (1) below. The geometric/analytic approach treated in this report introduces parabolic systems of partial dierential equations (PDEs) in a way which complements the more algebraic methods, which until now are better known in contact geometry. This is a report on joint work in progress with Hansjorg Geiges of the University of Leiden, Netherlands and Matthias Schwarz of the University of Leipzig, Germany. Many of the speci c results reported on here appeared rst in a paper [1] by Steve Altschuler, which introduced the heat- ow method to study contact structures, and in a recent preprint [2] of Altschuler and Lani Wu. 1. Introduction to Contact Geometry Many of the participants in this conference apply analytical methods to geometrically motivated problems, or use geometric methods to strengthen their analysis. However, it cannot be assumed that everyone is familiar with all of the most modern concepts and techniques of dierential geometry. For that reason, this section will be devoted to an introduction to contact geometry appropriate for analysts, among others, and may be skipped by those with a good knowledge of the area. I was until rather recently a complete novice in this area of geometry, and the reader should not expect a polished nor absolutely concise presentation. See [4], [5] and [6] for more complete references to the literature. I expect that analysts will be interested to see this novel application of parabolic operators. Date : September 29, 2000; November 24, 2000.

This work was initiated during my stay at the Max Planck Institute for Mathematics in the Sciences, Leipzig, and completed while I was visiting Monash University and the University of Melbourne. I would like to thank MPI, Monash and Melbourne for their generous hospitality. 106

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

107

A hyperplane distribution  in an open set M of R 2n+1 ; or in a smooth (2n + 1)-dimensional manifold M; speci es a subspace x of dimension 2n in R 2n+1 ; or rather in the tangent space to M at each point x 2 M; which depends smoothly on the point x: 1.1. Example: a foliation. A familiar example of a hyperplane distribution would be the two-dimensional distribution in R 3 spanned by the vector elds e1 (x) = (1; 0; 2x1) and e2 (x) = (0; 1; 2x2): Here we have written x = (x1; x2 ; x3 ): This distribution is especially easy to visualize, since e1 (x) and e2 (x) are a basis for tangent vectors to the family of paraboloids of revolution x3 , x12 , x22 = C; for various real constants C: This family of surfaces is a foliation of R 3 ; which means that every point of R3 lies on one of the surfaces, the surfaces and the family are smooth, and in some neighborhood of any point, the family looks like the family of coordinate planes x3 = const :; up to a local dieomorphism. In this situation, we say that the distribution is integrable, meaning in this case, where the rst and second components of e1 and e2 are (1; 0) and (0; 1); that their third components 2x1 and 2x2 are simultaneously the partial derivatives of a scalar function, locally. (The scalar function is x12 + x22 + C; of course.) Integrability is equivalent to saying that for any two vector elds V; W in ; the Lie bracket [V; W ] also lies in : Alternatively, we may describe a hyperplane distribution as the kernel of a nowhere vanishing dierential 1-form : (A 1-form is the dual of a vector eld, so that for any vector eld V; (V ) de nes a scalar function and depends linearly and pointwise on V:) Given ; the 1-form  is determined up to a nonvanishing scalar factor by the requirement that (e1) = (e2) = 0; where e1; e2 form a local basis for the distribution : (Computationally,  has the same components as the cross product of e1 and e2 .) The integrability condition for the distribution  may be written in terms of the 1-form  as an identity between 3-forms:  ^ d = 0: (The exterior derivative d of a 1-form  is the 2-form de ned by the alternating part of the matrix of rst partial derivatives; the wedge product of dierential forms is the alternating part of their tensor product.) A contact structure is a hyperplane distribution which is maximally non-integrable. In terms of Lie brackets, we may write ! (V; W ) for the transversal component ([V; W ]) of the Lie bracket of two vector elds V; W in : This makes ! a 2-form. The integrability condition requires that !  0; for  to be a contact structure, we require not merely that ! 6= 0 but far more: that the 2n-form ! n = ! ^ ! ^


^ ! be nowhere zero on  . Via the appropriate Riemannian metric, this is equivalent to saying that ! de nes an almost-complex structure on the hyperplane

108

ROBERT GULLIVER

distribution : Restricted to  = ker ; ! is the same as ,d: Thus, the contact criterion may be written entirely in terms of the 1-form  : (1)  ^ dn 6= 0: Note that inequality (1) depends on  but is independent of the choice of 1-form ; since if  = f for some nonvanishing scalar function f; then  ^ dn  f n  ^ dn: Note also that since d is a two-form, the left-hand side of (1) is a dierential form of degree 2n + 1; so on R n or M n ; it has only one real component. In this sense, contact structures and contact forms only have their full meaning in domains and manifolds of odd dimension. A 1-form  on a (2n + 1)-manifold which satis es inequality (1) is called a contact form. Inequality (1) is unusual, in the context of geometric analysis, for two reasons: it is a strict partial dierential inequality, and it is an underdetermined \system" consisting of one real, rst-order, fully nonlinear partial dierential inequality for the 2n +1 real components ai(x) of the 1-form : Speci cally, in the 5-dimensional case n = 2, we may write  in local coordinates (x ; : : : ; x ) as

e

e e

2 +1

+1

2 +1

X a (x) dx : = 0

4

4

X sgn() a

i

i=0

i

Then the contact inequality (1) is equivalent to the inequality (0)



@a(1) @a(2) 6 0; = @x(3) @x(4)

where the sum is over all permutations  of f0; 1; 2; 3; 4g: Systems of partial dierential equations of this general form are rather poorly understood at present. In the case of contact geometry, however, we shall see that there is a parabolic method available to attack inequality (1); see Section 2 below. 1.2. Example: the standard contact structure. A familiar example of a contact structure would be the two-plane distribution  in R with the subspace x at the point x = (x ; x ; x ) having basis vector elds e (x) = (x ; x ; 0) and e (x) = (,x ; x ; r ); where we have written r = x + x : In order to visualize ; we note that e is the horizontal vector eld pointing away from the x -axis, and that e is a vector orthogonal to e and with slope r; as measured from the (x ; x )plane. Then the distribution  is not a foliation, which may be seen as follows. Suppose (x (t); x (t)); 0  t  T; describes a closed curve in the (x ; x )-plane. Since x is never vertical, there is a unique way to lift this curve to a curve x(t) = (x (t); x (t); x (t)) in R ; so that 3

1

2

1

2 1

1 2 2

2

2

2

3

2

1

2

1

3

2

1

1

1

1

2

2

1

2

3

3

2

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

109

the tangent vector x0 (t) is always in the distribution x t : If  were integrable, then the space curve would stay on the same surface of the foliation, and would therefore be a closed curve. However, to be speci c, suppose that the plane curve (x (t); x (t)) describes the boundary @ ; in the positive sense, of = one-fourth of an annulus: in polar coordinates (r; ); is given by a < r < b; 0 <  < =2: Then along each of the two straight sides   0;   =2; the tangent vector lifts to a multiple of e = (x ; x ; 0), so x (t) remains constant. But along the quarter-circle r  b; 0 <  < =2; the tangent vector lifts to a multiple of e = (,x ; x ; r ), so x (t) increases by the slope times the length in the plane = b =2: Returning along the quarter-circle r  a; as  decreases from =2 to 0; x (t) decreases by a =2: Thus, the change in x (t) as t increases from 0 to T is  (b , a )=2; which is exactly twice the area of the quarter-annulus : In fact, for any domain in the (x ; x )-plane, the change in x (t) as (x (t); x (t)) describes @ equals twice the area of : This may be seen by computing a form  so that  = ker  :  = x dx , x dx + dx : Since x0(t) is in the distribution x t ; we get  (x0 (t)) = 0; which means that x0 (t) = ,x (t)x0 (t)+ x (t)x0 (t); hence the change x (T ) , x (0) in the height as (x (t); x (t)) goes around @ equals the integral around @ of ,x dx + x dx ; which is twice the area of : The 1-form  is the standard contact form on R ; and  is the standard contact structure. More precisely, this is the rotationally symmetric version; the contact form x dx + dx is translationally invariant in two coordinate directions, and is also known as \the" standard contact form. In higher dimensions, the standard contact form on R n is ( )

1

1

2

2

1

1

2

2

2

3

3

2

2

3

2

3

2

1

1

2

3

2

0

0

0

2

1

1

2

( )

2

3

1

2

1

1

1

0

3

2

3

2

1

2

3

0

2

1

X  = dx + (x n

(2)

3

0

2k

0

k=1

3

2 +1

dx2k,1 , x2k,1 dx2k );

which is invariant under the (n + 1)-dimensional group generated by rotation in the (x k, ; x k )-plane, 1  k  n; plus translation along the x -axis. 2

1

2

0

A natural question is: when are two contact forms equivalent? The local version of this question has a surprisingly simple answer: Theorem 1.1. (Darboux): Let  be a contact form on a neighborhood of x in R n : Then on a smaller neighborhood of x, there is a dieomorphism into R n such that  is mapped to the standard contact form  : 2 +1

2 +1

0

110

ROBERT GULLIVER

Darboux' Theorem may be interpreted as saying that the contact condition (1) is a very \soft" condition, as compared to the familiar partial dierential equations traditionally treated by geometric analysts. This softness is apparent from the recent work of Gromov, Eliashberg and others on noncompact manifolds, which showed, for example, that any noncompact, odd-dimensional manifold which has a hyperplane distribution with an almost-complex structure also carries a contact structure (see [8] and references therein.) 1.3. Global non-uniqueness: the Lutz Twist. Since contact structures are locally unique, it might seem reasonable to think that a topologically simple space like R 3 has only one contact structure up to a change of coordinates. However, there are subtle criteria which distinguish other contact structures on R3 from the standard 0: Recall the description in subsection 1.2 of basis vector elds e1 ; e2 for the standard contact structure on R 3 : e2 is orthogonal to the radial vector e1 ; and has slope r; which means that it makes an angle ' = arctan r with the (x1 ; x2 )-plane. As r ! 1; e2 becomes vertical, so ' ! =2: Instead, suppose that ' = '(r) increases beyond =2 to make one or more revolutions before slowly approaching arctan r +2m (m 2 Z) as r ! R < 1: Outside the cylinder r < R; the contact structure may be continued smoothly, to join up with the standard contact structure. This construction is known as the Lutz twist (see [9].) In terms of the contact form, in cylindrical coordinates (r; ; x3); 0 = dx3 , r2 d is replaced by  = h0 (r)dx3 , h1(r) d for some functions h0 ; h1 : [0; 1) ! R with h1 h0 , h1 h0 > 0; and with h0(r) = 1; h1(r) = r2 for all r  R: Then h1 and h0 are related to the angle ' by rh0(r) tan '(r) = h1(r): This new contact structure is overtwisted, that is, there is a topological disk D  R 3 with jD nowhere zero along @D and j@D  0: In fact, let r0 be the rst value of r with '(r0) = : Then we may choose D = f(r; ; x3) : x3 = r02 , r2 ; 0    2; 0  r  r0g: It may be shown that no such disk exists in R 3 with the standard contact structure. 1.4. Compact Manifolds. What about compact manifolds? The only known obstruction to the existence of an orientable contact structure on an oriented, odd-dimensional manifold M 2n+1 is the requirement that some hyperplane distribution on M should have an almost-complex structure; this can be written as a topological condition on M , that the even-dimensional Stiefel-Whitney classes w2i (certain natural cohomology classes with Z=2Z coecients) are in the image of cohomology with integer coecients. However, there are many manifolds which satisfy 0

0

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

111

this condition but have not been shown to carry a contact structure. Speci cally, one would like to know whether there is a contact structure on the odd-dimensional torus T 2 +1 . We shall assume for the rest of this paper that manifolds are compact, oriented and have no boundary. A readily visualized example is the interesting case of the torus T 2 +1, which is just the cube [,; ]2 +1  R 2 +1 after opposite faces have been identi ed. A contact form may be found on the three-torus T 3 as the rst case of a classical construction. Begin on the two-dimensional torus T 2; introduce local coordinates (q0 ; q1) on T 2 and then extend these coordinates to the 4-dimensional phase space, or cotangent bundle, T (T 2): Then a cotangent vector  at the point x = (q0 ; q1) has components (p0; p1), meaning that  = p0 dq0 + p1 dq1: (In certain applications, (q0; q1 ) are coordinates of position and (p0; p1 ) are components of the momentum vector.) Then ! = dp0 ^ dq0 + dp1 ^ dq1 is the natural symplectic form on phase space T (T 2): One notes that ! is the exterior derivative d; where  is the canonical 1-form p0 dq0 + p1 dq1 on phase space. When  is restricted to the unit-sphere bundle M 3 := f(x; p) : x 2 T 2; p 2 T (T 2); jpj2 = 1g, it satis es the contact condition (1). Here jpj2 = p20 + p21: The veri cation of inequality (1) reduces to showing that p0 @ jpj2 =@p0 + p1 @ jpj2 =@p1 6= 0 on M . Meanwhile, on T 2; there is a global basis of tangent vector elds, which implies that the unit sphere bundle M 3 of T 2 is T 2  S 1 = T 3: In coordinates (q0; q1 ; ) for T 3 = (R =2Z)3 ; we have  = cos  dq0 + sin  dq1 : This is the most natural construction for a contact structure on T 3: The construction above generalizes to higher dimensions. Let N be an oriented (n +1)-dimensional manifold, equipped with a Riemannian metric, and introduce local coordinates (q0 ; : : : ; q ; p0; : : : ; p ) for the cotangent bundle T N of N; where (q0 ; : : : ; q P ) are local coordinates on N and a cotangent P vector is represented as =0 p dq : Let  be the canonical 1-form =0 p dq : When  is restricted to the unit-sphere bundle M 2 +1 ; de ned as (q; p) : q 2 N; p 2 T N; jpj2 = 1 ; it satis es the contact condition (1). That is, the unit sphere of the cotangent bundle of any manifold carries a natural contact structure. This is how contact structures arise naturally, on suitable energy surfaces in Hamiltonian systems. When one applies the same construction to N = T 3; n = 2; one nds a contact 1-form 1 on the 5-dimensional unit sphere bundle of T N: But the unit sphere bundle M 5 is now T 3  S 2; not T 5 : However, T 5 can still be given a contact structure, as was rst shown by Lutz [9]. Another way to nd a contact structure on T 5 is to apply the following result of Gromov (see [8] and [5], p. 456): n

n

n

n

x

x

x

n

n

n i

n

n

i

i

n

i

q

i

i

112

ROBERT GULLIVER

Theorem 1.2. : If M2 is a branched covering of M1 ; branched along a codimension-2 submanifold  of M1 ; and if M1 has a contact form 1 whose restriction to  also makes  into a contact manifold, then M2 has a contact form  close to the pullback of 1 : In our case, T 2 may be written as a branched double cover F : T 2 ! S 2 of the sphere, branched simply over four points of S 2 ; which we may assume are the four equidistant points (1; 0; 0); (0; 1; 0) along the equator fp2 = 0g of S 2  R 3 : We construct a branched covering Fe : M2 ! M1 from M2 = T 5 = 3 T T 2 to M1 = T 3 S 2 ; by twisting F; as follows. Let q = (q0 ; q1 ; q2 ) be coordinates for T 3 = (R =2Z)3 ; and let p = (p0; p1 ; p2) be coordinates for S 2; where p02 +p12 +p22 = 1: For each q2 2 R =2Z; let (q2 ) : S 2 ! S 2 be the rotation in the (p0; p1)-plane by angle q2 ; leaving p2 xed. Then Fe : M2 ! M1 is de ned by Fe(q0 ; q1 ; q2 ; z ) := (q0 ; q1 ; q2 ; (q2 )(F (z ))) : Fe : M2 ! M1 is a branched covering, with branch locus  = f(q; p) 2 T 3  S 2 : p0 = cos(q2 + k=2); p1 = sin(q2 + k=2); p2 = 0; k 2 Zg:  has four connected components  ; k = 0; 1; 2; 3; each of which projects dieomorphically onto the the T 3 factor of M1 : P 2 Write 1 for the canonical contact 1-form =0 p dq on M1 ; viewed as the unit cotangent bundle of T 3: On each component  of ; we have 1 jk = cos(q2 + k=2) dq0 + sin(q2 + k=2) dq1: We compute (1 ^ d1) jk = ,dq0 ^ dq1 ^ dq2 ; k = 0; 1; 2; 3; which shows that  is a (disconnected) contact 3-manifold with contact form 1 j : We may now apply Theorem 1.2 to nd a contact form  on M2 = T 5 which is close to the pullback of 1 : Thus, the 5-torus T 5 has a contact structure. The existence of a contact structure on the 7-torus, and on numerous higher-dimensional manifolds, was unclear until now. k

i

i

i

k

2. The Heat Flow Recall that we are assuming that manifolds are compact, connected, oriented and have no boundary. In addition, we will assume that a Riemannian metric has been chosen. A property of parabolic PDEs familiar to analysts is the strong maximum principle: if the solution f (t; x) satis es at the initial time f (0; x)  0 but f (0; x) 6 0; then at time t > 0; f (t; x) will be positive everywhere. That is, heat ows instantaneously to warm a connected domain. This property makes parabolic methods ideal for the study of strict inequalities such as the contact inequality (1). The idea is to

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

113

use a hands-on construction to make f (0; x) > 0 for x in an appropriate, possibly quite small, open set, while f (0;
)  0 everywhere, and then to replace f (0; x) with the strictly positive solution f (t; x) at some small positive time t: Since f (t; x) is close to f (0; x) in certain strong norms, other relevant conditions will be maintained. Altschuler in [1] considers any orientable compact 3-manifold M : he combines the Lutz twist with the strong maximum principle to construct a contact form on M: (The result was proved using entirely dierent methods in [10]; see also [4].) Altschuler's technique is to start with a foliation, or equivalently with a 1-form 1 satisfying 1 ^ d1  0; and then to use the Lutz twist to construct a 1-form 2 satisfying 2 ^ d2 > 0 on a certain open set U; with 2 = 1 near @U: The resulting 1-form 2 on all of M satis es 2 ^ d2  0; such 1-forms have been called confoliations by Eliashberg and Thurston [4]. Altschuler then de nes a degenerate parabolic system of equations for a 1-form (t; x); 0 < t < "; x 2 M; and uses 2 as the initial condition at time t = 0: The system of PDEs is chosen so that the scalar quantity f (t; x) := ( ^ d); which is initially nonnegative everywhere and strictly positive on U; becomes everywhere positive for small t > 0: One diculty is that the system of PDEs is degenerate parabolic, so that \heat" will ow reliably only in certain directions. Altschuler de nes the system of equations so that heat ows in directions tangent to ker 2 ; which is the original foliation ker 1 on the more troublesome set M nU; and ensures that the Lutz twist was carried out so that the open set U meets each leaf of ker 1: A nonlinear version of the system of equations Altschuler uses on a 3-manifold is @ (3) =  ( ^ df ) ; where f (t; x) = ( ^ d): @t

Here, for a p-form on an oriented Riemannian (2n +1)-manifold,  is a (2n+1,p)-form, the Hodge star of ; which depends linearly on and is de ned at each point so that for any oriented orthonormal coframe 0 ; : : : ; 2n of 1-forms, (p ^ : : : ^ 2n ) = 0 ^ : : : ^ p,1 : The system (3) appears quite complicated, but it may be dealt with successfully by the following trick. The real-valued function f (t; x) satis es a single degenerate parabolic PDE: (4)

@f =  ( ^ d  ( ^ df )) + h ^ df; di: @t

Thus, the system (3) uncouples weakly, in the sense that  appears in the PDE (4) only as a coecient. Once f (t; x) is determined, the equation (3) for (t; x) becomes a parameterized system of ODEs. Of

114

ROBERT GULLIVER

course, the unknown 1-form  also appears in the coecients of (4), so this version of Altschuler's method succeeds by requiring t to remain small, implying that (t;
) is close to the initial 1-form 2 : More generally, on a (2n + 1)-manifold, choose a (2n , 1)-form , and consider the system of equations

@ =  ( ^ df ) ; where f (t; x) := ( ^ d): @t The PDE satis ed by f is now   @

@f (6) @t =  ( ^ d  ( ^ df )) +  @t ^ d : (5)

Again, the system (5) uncouples weakly. Further, we have Proposition 2.1. : Equation (6) is a weakly parabolic PDE, a degenerate heat equation, where the right-hand side de nes a second-order partial dierential operator, which is strongly elliptic when restricted to the distribution H  TM given by

H = (ker( ))? : Recall that for a 2-form on M; ker x := f v 2 TxM j (v;
) = 0 on TxM g. The coecients of the principal part of the PDE (6) at (t; x) are AT A; where the skew-symmetric matrix A represents  x in coordinates which are orthonormal at x; the subspace Hx is spanned by the columns of AT A:

In the nonlinear version of Altschuler's heat ow on a 3-manifold, as we have seen, one chooses = , the evolving 1-form itself. We would like to apply Proposition 2.1 in this case. At a given small time t > 0; the 1-form (t;
) never vanishes, so we may complete to a local orthonormal basis (1 ; 2 ; 3) with  a nonvanishing scalar multiple of 1: We compute ker() = ker(1 ) = ker(2 ^ 3 ) = R e1 ; and therefore H = (ker())? = R e2 + R e3 = ker : In particular, for small positive t; the distribution H is close to ker 2: Thus, if 2 is a contact form on an open set U which is a neighborhood of some point on each leaf of the original foliation ker 1, then heat will ow out of U to warm each point of M: In general, one may show that Lemma 2.2. : If is locally decomposable as a product of 1-forms, then H := (ker( ))? = ker :

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

115

3. The Higher Lutz Twist For higher dimensions 2n + 1, in the recent paper of Altschuler and Wu [2], the degenerate parabolic system (5) is studied with the choice

=  ^ (d)n, : They show the existence of a smooth solution for all positive time, via a parabolic regularization. (More precisely, in [2] Altschuler and Wu consider a partly linearized degenerate parabolic system which is easier to analyze than equation (5), and slightly more complicated, but has consequences equivalent to those of equation (5).) The PDE (6) now becomes 1

@f = n  ( ^ d  ( ^ df )) + h ^ df; (d)ni: @t

(7)

Thus, the system of equations uncouples in the same sense as in the 3-dimensional case n = 1 (compare equations (3) and (4).) Another part of their paper carries out a higher analogue of the Lutz twist for the ve-dimensional product case M = N  F ; using a contact structure on the 3-dimensional manifold N and its parallelizability. They are thereby able to prove that every product 5-manifold of this form carries a contact structure. Incidentally, this gives another construction of a contact form on the 5-torus T : Let us proceed in an analogous, but in applications rather dierent, fashion. Consider a (2n +1)-manifold M n = N n,  F which is the product of a contact (2n , 1)-manifold (N; N ) and an oriented surface F . We shall write  for the 1-form on M = N  F pulled back from N : For simplicity, assume that (N; N ) has a closed Reeb orbit . This means that 0 (s) 6= 0 and that d( 0(s); v) = 0 for all parameter values s along the curve and for all vectors v 2 T s M: Then, according to an extension of Darboux' Theorem 1.1, in some neighborhood W of in N , there are multipolar coordinates (z; r ;  ; : : : ; rn, ; n, ); r + : : : + rn, < R ; k 2 R mod 2; so that N is the standard contact form (2), which in these coordinates means that 5

3

2

5

2 +1

2

1

2

1

( )

2 1

2

1

2

1

N = dz +

1

1

1

X r d : n,1 k=1

2

k

k

In a small ball B  F ; let polar coordinates (rn; n ) be chosen, 0  rn < R; n 2 R mod 2: For some choice of real-valued functions hk (r ; : : : ; rn ), 0  k  n; de ne 2

1

(8)

X  = h (r ; : : : ; r ) dz + h (r ; : : : ; r ) d : n

2

0

1

k

n

k=1

1

n

k

116

ROBERT GULLIVER

Then 2 will satisfy the contact inequality (1) in U := W  B provided (9)

1

r1
: : :
rn





h0 h1 : : : hn @1 h0 @1 h1 : : : @1 hn

...

...

...



> 0; for ri  0 :

@n h0 @n h1 : : : @n hn Here the operator @k denotes @=@rk : Inequality (9) is equivalent to

the orientation preserving local dieomorphism property for the central projection of (h0 ; : : : ; hn) 2 R n+1 to the sphere S n: Observe that inequality (9) continues to hold when 2 is multiplied by a positive scalar function. Recall that we wish to carry out this higher Lutz twist on the open set U , but we need to construct 2 on all of M . Therefore it will be necessary for the coecients hk (r1; : : : ; rn) to satisfy boundary conditions on @U , so that the extension of 2 to all of M by de ning 2 = 1 on M nU will be smooth. However, only the oriented contact structure is important to us, which means that 2 only needs to be de ned modulo a (nonconstant) positive multiple. Speci cally, there needs to hold on the boundary hk (r1 ; : : : ; rn) = rk2 h0 (r1; : : : ; rn); 1  k  n , 1; and hn(r1; : : : ; rn) = 0; as well as inequality (9) in the interior of U: This requires us to nd a mapping from the sector V := f(r1 ; : : : ; rn ) 2 (0; 1)n : r12 + : : : + rn2 < R2 g to the sphere S n which is a dieomorphism of V with an open subset of S n; having the following boundary values on @V . On the curved part of the boundary fr12 + : : : + rn2 = R2g; we require hk = rk2 h0 ; 1  k  n , 1; and hn = 0: For 1  k  n; on the face frk = 0g; we require hk = 0; 1  k  n: In the ve-dimensional case n = 2; this may be done using a conformal mapping from the quarter-disk V to the hemisphere of S 2 with a slit from an interior point to the equator removed. The boundary of the quarter-disk covers the slit twice and the equator once. For the general case n  2, another more hands-on construction of the map from V into the hemisphere of S n may be carried out. Closed Reeb orbits may be rare for a given contact manifold (N; N ), but the above procedure may be modi ed appropriately. A covering argument may then be used to arrange disjoint open sets of M of the above form so that their projections from M = N  F to N cover all of N: For small time t > 0; the solution (t;
) will be close to the initial value 2: On the complement of the union of the sets U where the higher Lutz twist has been carried out, we have 2 = 1 : Write 1 = 1 ^ (d1 )n,1 : Since 1 is the pullback of the contact form N ; we see that 1 is the pullback of a volume form on N; and thus is decomposable as a product of 1-forms. It follows from

THE HEAT-FLOW METHOD IN CONTACT GEOMETRY

117

Lemma 2.2 that on the complement of the sets U; the distribution H1 of elliptic directions for the system (5), with replaced by 1 ; equals ker 1 ; which is the tangent plane T F to fyg F 2 in T M = T N  T F: Therefore, for small time t > 0; the distribution H is close to the foliation T F: It follows by Proposition 2.1, for small time t > 0; that heat ows out of the union of open sets U along directions arbitrarily close to T F to warm all of M = N  F: Numerous points omitted here, in part rather technical, will be treated in [7] to prove Theorem 3.1. : If N 2n,1 is a compact contact manifold and F 2 is a compact oriented surface, then M 2n+1 = N  F has a contact 1-form, which is C 2 -close to a 1-form 2 obtained from the contact form of N by means of the higher Lutz twist. By induction on n = 2; 3; : : : ; with N = T 2n,1 and F = T 2; we deduce Corollary 3.2. : Any odd-dimensional torus T 2n+1 carries a contact structure.

References

[1] S.J. Altschuler, A geometric heat ow for one-forms on three dimensional manifolds, Illinois J. Math. 39 (1995), 98{118. [2] S.J. Altschuler and L.F. Wu, On deforming confoliations, preprint 1999. [3] S.-S. Chern, Pseudo-groupes continus in nis in Geometrie Dierentielle, Colloques Internat. CNRS, Strasbourg 1953. [4] Y. Eliashberg and W. Thurston, Confoliations, Univ. Lecture Ser. 13, Amer. Math. Soc., Providence, 1998. [5] H. Geiges, Constructions of contact manifolds, Proc. Camb. Phil. Soc. 121 (1997), 455{464. [6] H. Geiges, Contact topology in dimension greater than three, to appear in Proc. Eur. Congr. Math. (Barcelona, 2000). Progress in Math., Birkhauser, Basel. [7] H. Geiges, R. Gulliver and M. Schwarz, Heat ow and contact structures on product manifolds, in preparation. [8] M. Gromov, Partial Dierential Relations, Ergebnisse vol. 9, Springer, Berlin 1986. [9] R. Lutz, Sur la geometrie des structures de contact invariants, Ann. Inst. Fourier (Grenoble) 29 (1979), 283{306. [10] J. Martinet, Formes de contact sur les varietes de dimension 3, Lecture Notes in Math. 209 (1971), 142{163: Springer, Berlin. School of Mathematics, University of Minnesota

E-mail address :

[email protected]

MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV

Abstract. The description of electron current through a splitting is a mathematical problem of electron transport in quantum networks [5, 1]. For quantum networks constructed on the interface of narrow-gap semiconductors [29, 2] the relevant scattering problem for the multi-dimensional Schodinger equation may be substituted by the corresponding problem on a one-dimensional linear graph with proper selfadjoint boundary conditions at the nodes [11, 10, 25, 24, 16, 19, 4, 28, 20, 18, 6, 5, 1]. However, realistic boundary conditions for splittings have not yet been derived. Here we consider some compact domain attached to a few semiin nite lines as a model for a quantum network. An asymptotic formula for the scattering matrix for this object is derived in terms of the properties of the compact domain. This allows us to propose designs for devices for manipulating quantum current through a splitting [3, 15, 22, 9, 21].

Introduction: current manipulation in the resonance case

In this paper we discuss the scattering problem on a compact domain with a few semi-in nite wires attached. This is motivated by the design of quantum electronic devices for triadic logic. In the papers [3, 15] a special design of the one-dimensional graph which permits manipulation of the current through an elementary ring-like splitting is suggested. This permits, in principle, manipulation of quantum current in the resonance case to form a quantum switch. Another device for manipulating quantum current through splittings is discussed in [22, 9]. In [21] the special design of the splitting formed as a circular domain with four one-dimensional wires attached is used to produce a triadic relay. Date : 20 September 2000 Revised 20 January 2001.

118

MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 119

In order to illustrate the basic principle of operation consider the selfadjoint Schrodinger operator  L  ,
+ q(x) @ @n @ = 0: on some compact domain . In this paper we will only consider the case  R 3 [22] (for other cases see also [3, 15, 21]). Roughly speaking the solution of the Cauchy problem  @ = L i@t (1) (x; 0) = 0(x) is given in terms of eigenfunctions 'n X (x; t) = neint 'n(x): n

Picking a speci c mode '0 with energy 0 we suppose that '0 disappears on some subset l0  . Connecting `thin channels' at various

λ0

λ0

Ω

l0

gure 1. Resonance switch points on the boundary of and introducing an excitation of energy 0 along the channels we can hope to create a switching eect. Essentialy this is achieved by varying q(x) so that l0 \ @ coincides with the connection point of a `thin channel'. Implicit in our construction is the assumption that the energy of the electrons in the device is equal to some resonance eigenvalue of the Schrodinger operator on . We refer to this as the resonance case1. This has interesting implications when we consider the eect of decreasing the length scale|or equivalently scaling up the energy|viz. the eect of non-zero temperature becomes negligable for suciently small length scales, see [14]. 1

120

M. HARMER, A. MIKHAILOVA AND B. S. PAVLOV

Another assumption which we have made above is that 0 is a simple eigenvalue of L. We will show that the case of multiple eigenvalues is a simple generalisation of the case for simple eigenvalues, see [3, 15]. In the rst section we give a brief description of the connection of the thin channels (here they are modelled by one-dimensional semi-lines) to the compact domain, for more details see [22]. In the second section we derive an asymptotic formula for the scattering matrix in terms of the eigenfunctions on the compact domain. In the last section we brie y discuss some simple models of a quantum switch constructed on the basis of this asymptotic formula. 1. Connection of compact domain to thin channels As we mentioned above the thin channels are modelled by onedimensional semi-lines. This is justi ed by an appropriate choice of materials (narrow-band semiconductors) and energies, see [29, 2, 21]. We assume that these channels are attached at the points fa1 ; a2 ; :::; aN g  @ (perturbation of the operator L at inner points faN +1 ; aN +2 ; : : : ; aN +M g may be considered using the same techniques as for fa1 ; a2 ; :::; aN g [2, 3, 21] although we do not consider this here). We refer to L, de ned above, as the unperturbed Schrodinger operator. L is restricted to the symmetric operator L0 de ned on the class D0 of smooth functions with Neumann boundary conditions which vanish near the points a1 ; a2; : : : ; aN . The de ciency subspaces, Ni, of the restricted symmetric operator L0 , [L?0  i] esi = 0 for complex values of the spectral parameter  coincide with Greens functions G(x; as) of L which are elements of L2 ( ) but do not belong to the Sobolev class W21 ( ). In the case when is a compact one-dimensional manifold (a compact graph) these Greens functions are continuous and can be written in terms of a convergent spectral series [3]. However, when  R 2 ; R 3 the de ciency elements will have singularities and we must use an iterated Hilbert identity to regularise the values of the Greens function at the poles. It is well known, for  R 3 , that the Greens function admitts the representation inside

(2)

p ei jx,yj 0 G (x; y ) = + g(x; y; ) 4jx , yj

MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 121

where g(x; y; ) is continuous. The potential-theory approach gives the asymptotics of the Green function near the boundary point as 2 @

G0 (x; as)  2jx1, a j + Ls(x) + Bs(x; ); (3) s where Ls is a logarithmic term depending only on @ and Bs is a bounded term containing spectral information [8]. In order to choose regularised boundary values we use the following lemma [22] (here we assume L > ,1 is semi-bounded from below): Lemma 1. For any regular point N+from the complement of the specM trum  (L) of L and any a 2 fasgs=1 the following representation is true: G(x; a) = G,1(x; a) + ( + 1)G,1  G (x; a); where the second addend is a continuous function of x and the spectral series of it in terms of eigenfunctions 'l of the nonperturbed operator

L

( + 1)G,1  G(x; a) = ( + 1)

X l

'l (x)'l (a) (l + 1)(l , )

is absolutely and uniformly convergent in . The proof of this lemma is based on the classical Mercer theorem along with the Hilbert identity [22]. It is well known that the domain of L?0 can be written as the direct sum (4) D0? = D0 + Ni + N,i so for any u 2 D0? X X u = u0 + A+s Gi(x; as) + A,s G,i(x; as): s

s

We de ne u 2 D0 in terms of the coordinates As  A+s +"A,s ; ?

u(x) , Bs  xlim !a s

X t

#

At 0. The resulting self-adjoint extension we denote by L . The parameter is a measure of the strength of the connection between the rays and the compact domain|in the limit ! 0 the resolvent of L converges uniformly to the resolvent of L on each compact subset of the resolvent set of L [22]. 0

2. Asymptotics of the scattering matrix For the remainder we assume that the potential on the rays qs(xs)  0 is zero. We use the ansatz (8) us = fs(xs; ,k)s1 + fs(xs; k)Ss1;

MANIPULATING THE ELECTRON CURRENT THROUGH A SPLITTING 123

for the scattered wave generated by the incoming wave from the ray attached to the point a1. Here fs(xs; k) are the Jost solutions [27], in this case (qs(xs)  0) just the exponentials fs(xs; k) = eikxs ; and  = k2 is the spectral parameter. From the boundary conditions (7) we get N equations As = fs0 (0; ,ik)s1 + fs0 (0; ik)Ss1 (9) Bs = fs(0; ,ik)s1 + fs(0; ik)Ss1: Inside the eigenfunction u(x; k) may be written as a sum of Greens functions at the spectral parameter  = k2

u(x; k) =

N X s

CsG (x; as):

Using the Cayley transform between the spectral points i and  one gets a relationship between these Greens functions and the de ciency elements (as de ned above) so that [22]  I + L  lim [G(x; a ; ) , 0, but not for  = 0. More generally, if we put log  = log ; logn  = log(logn  ) we have the iterated logarithm functions. Then for each n > 0 the function 1

+1



log (1 +  ) log (e +  ) log (ee +  )


(logn(ee +  )) again satis es (4) for all  > 0, but not for  = 0.




1

2

e

1+

3

Finally, (3) is no accident. We can prove that if eA K 2 Lploc(B) for some p  0 and certain types of A (and in particular the log log : : : examples), with (

Z

B

)

A(K (z)) dz = 1

e

then there is a homeomorphic (and hence continuous) solution in W ;p(B) for all p < 2. In fact the solution lies in an Orlicz{Sobolev class just below Wloc; 1

12

4. Mappings of Finite Distortion We next give a general de nition of the mappings which mostly occur. Roughly, solutions to a Beltrami equation with ellipticity bounds which are pointwise nite will be mappings of nite distortion as soon as they are ACL{absolutely continuous on lines. De nition A mapping f : ! C is said to have nite distortion if:

1. f 2 Wloc; ( ), 2. The Jacobian determinant, J (z; f ) = det Df (z), of f is locally integrable and does not change sign in

3. There is a measurable function K = K(z)  1, nite almost everywhere, such that f satis es the distortion inequality (8) jDf (z)j  K(z) jJ (z; f )j a:e:

Notice that the hypotheses are not sucient to guarantee that f 2 ; Wloc ( ) unless the distortion function K is bounded. Nor do they imply that the Jacobian does not vanish on a set of positive measure. 11

2

12

The motivational philosophy behind the condition that the distortion function is exponentially integrable is now clear. We wish to exploit the BM O , H duality (and even more re ned versions of this) to achieve uniform estimates on approximating sequences of solutions. 1

136

TADEUSZ IWANIEC AND GAVEN MARTIN

5. Maximum Principle and Continuity One of the rst tasks is to establish a maximum type principle and modulus of continuity estimates. A continuous function u : ! R de ned in a domain is monotone if oscB u  osc B u for every ball B  . This de nition in fact goes back to Lebesgue in 1907 where he rst showed the relevance of the notion of monotonicity to elliptic PDEs in the plane. In order to handle very weak solutions of dierential inequalities, such as the distortion inequality, we need to extend this concept, dropping the assumption of continuity, and to the setting of Orlicz{Sobolev spaces. De nition. A real valued function u 2 W 1 ( ) is said to be weakly monotone if for every ball B  and all constants m  M such that (9) jM , uj , ju , mj + 2u , m , M 2 W01 (B) we have (10) m  u(x)  M for almost every x 2 B. For continuous functions (9) holds if and only if m  u(x)  M on @ B. Then (10) says we want the same condition in B, that is the maximum and minimum principles. Here, and in what follows we assume, unless otherwise stated, that the Orlicz function P satis es Z1 P (t) dt (11) t3 = 1 1 and that the function t 7! t 58 is convex. For example Orlicz functions of the form @

;P

;P

2 e log1 (1 +  ) log2 (e +  ) log3(e +  )


(log e +  ) are of this form. The Orlicz{Sobolev space W 1 ( ) consists of functions which, together with their rst derivatives, lie in the space L ( ). Thus in the example given above we are looking in Zygmund type spaces just below W 1 2( ). P ( ) =

e

n

;P

loc

P

;

loc

e






WHAT'S NEW FOR THE BELTRAMI EQUATION ?

137

Lemma 5.1. Let be a bounded domain and suppose that u 2 W 1;P ( )\ C ( ) is weakly monotone. Then (12) min u  u(x)  max u @

@

for every x 2 . The paper [17] by Manfredi should be mentioned as the beginning of the systematic study of weakly monotone functions. We now recall a fundamental monotonicity result in the Orlicz{ Sobolev classes. Theorem 5.1. The coordinate functions of mappings with nite distortion in W 1;P ( ) are weakly monotone. There is a particularly elegant geometric approach to the continuity estimates of monotone functions. The idea goes back to Gehring in his study of the Liouville theorem in space where he developed the Oscillation Lemma. We need the P {modulus of continuity P ( ) de ned for 0   < 1 as follows. For  > 0 the value t of P at  is uniquely determined by the equation Z 1=

(13)

1

ds P (st) 3 = P (1): s

Certainly P is a non-decreasing function with (14) lim  ( ) = 0:  !0 P Given the transcendental nature of the equation one must solve, it is impossible in all but the most elementary situations, to calculate . Here are a few explicit formulas for ( ) which exhibit the correct asymptotics for  near 0. 1 P (t) = t2 ; ( ) = j log  j, 2 More generally for all  > 0 we have,  P (t) = t2 log,1 (e + t);  > 0; ( )  j log  j, 2

and nally P (t) =

1 t2 ; ( )  [log j log  j], 2 ; P (t) = log(e + t)

t2

; ( )  [log log j log  j], 2 ; 1

log(e + t) log log(3 + t) We now have the fundamental modulus of continuity estimate.

138

TADEUSZ IWANIEC AND GAVEN MARTIN

Theorem 5.2. Let u 2 W 1 (B) be weakly monotone in B = B(z0 ; 2R) . Then for all Lebesgue points a; b 2 B(z0 ; R) we have  ja , bj  ju(a) , u(b)j  16RkrukB  2R : (15) ;P

;P

In particular, u has a continuous representative for which (15) holds for all a and b in the disk B(z0 ; R). In the statement above we   have used Z 1 1 (16) P (jruj)  P (1) krukB = inf :  jBj B to denote the P -average of ru over the ball B. ;P

Theorem 5.3. 1Every mapping with nite distortion in the Orlicz{ Sobolev class W ( ), is continuous. ;P

loc

6. Liouville Type Theorem Here is a rst taste of the power of Theorem 5.1. Theorem 6.1. Let f : C ! C be a mapping of nite distortion whose dierential belongs to L (C ). Then f is constant. The proof consists in showing that R krukB ! 0 as R ! 1 in (15) using the Dominated Convergence Theorem. P

;P

7. Solutions A principal solution is a homeomorphism h : C ! C with 1. a discrete set E (the singular set) such that h 2 W 1 1(C n E ), 2. the Beltrami equation h (z) = (z)h (z) holds for a.e. z 2 C , and 3. we have the normalisation h(z) = z + o(1) at 1 It will become clear that the key to understanding the Beltrami equation and its local solutions is in the existence and uniqueness properties of the principal solutions. A function f , not necessarily a homeomorphism, is a very weak solution if it satis es:  there is a discrete set E  (the singular set) such that h 2 W 1 1 ( n E ).  the Beltrami equation f (z ) = (z )f (z ) holds for almost every z 2

;

loc

z

z

f

;

loc

f

z

z

WHAT'S NEW FOR THE BELTRAMI EQUATION ?

139

The point is that one expects a solution f to be the composition of the principal solution h with a meromorphic function ' de ned on h( ), f = '  h; (the Stoilow factorisation theorem). Thus away from the poles, the solution f is locally as good as the principal solution. 7.1. Uniqueness of Principal Solutions. Here is the most general uniqueness result that we are aware of. Theorem 7.1. Every elliptic equation h = H (z; h ) admits at most one principal solution in the Sobolev-Orlicz class z + W 1 (C ) We use the term z + W 1 (C ) to denote the mappings h with jh j + jh , 1j 2 L (C ). As far as the ellipticity is concerned, we assume that there is a measurable compactly supported function k : C ! B such that for almost every z 2 C and all ;  2 C jH (z;  ) , H (z;  )j  k(z)j ,  j z

z

;P

;P

z

P

z

Proof. Let h be a solution to the equation. Thus h (z) = 0 for suciently large. The point is that given two principal solutions h1 and h2 , the mapping f = h1 , h2 has nite distortion and its dierential Df = Dh1 , Dh2 belongs to L (C ). To see this note j(h1 , h2) j = jh1 , h2 j = jH (z; h1 ) , H (z; h2)j  k(z)j(h1 , h2 ) j whence J (z; h1 ,h2 )  0. It follows that f is constant from the Liouville theorem. The normalisation at 1 implies that this constant is 0. z

z

P

z

z

z

z

z

z

8. Stoilow Factorisation We now state that if a Beltrami equation admits a homeomorphic solution, then all other solutions in the same class are obtained from this solution via composition with a holomorphic mapping. Theorem 8.1. Suppose we are given a homeomorphic solution h 2 1 W ( ) to the Beltrami equation h = ( z ) h (17) a:e:

;P

loc

z

z

140

TADEUSZ IWANIEC AND GAVEN MARTIN

Then every solution f 2 W 1 ( ) takes the form f (z ) = (h(z )); z 2

where  : h( ) ! C is holomorphic. ;P

loc

9. Failure of Factorisation Now an example which shows that for fairly nice solutions one cannot expect a factorisation theorem even in the case of bounded distortion. Theorem 9.1. Let K > 1 and q0 < 2+1 . Then there is a Beltrami coecient  supported in the unit disk with the following properties. ,1 ,  kk1 = +1 continuous so The Beltrami equation h = h admits a Holder lution h 2 z + W 1 0 (C ) which fails to be in W 1 2 (C ),  The solution h is not quasiregular, and therefore not the principal solution, nor obtained from the principal solution by factorisation. K

K

K

K

z

z

;

;q

loc

10. Distortion in the Exponential Class Theorem 10.1. There exists a number p0 > 1 such that every Beltrami equation h (z ) = (z ) h (z ) a:e: C with Beltrami coecient  such that (z ) , 1  (z ) j ( z ) j  K K (z ) + 1 B and (18) e 2 L (B) with p  p0 , admits a unique principal solution h 2 z + W 1 2 (C ). There are examples to show that in order for there to be a principal solution in the natural Sobolev space z + W 1 2(C ) it is necessary that the exponent p at (18) is large, at least p  1. As a matter of fact, somewhat more is true in Theorem 10.1. The higher the exponent of integrability of e the better the regularity of the solution. That is even beyond L2 , such as L2 log L with any   0, see [16]. z

z

K

p

;

;

pK

The situation is dierent if the integrability exponent of e is smaller than the critical exponent p0. Here the principal solution need not be in z + W 1 2(C ), but we still obtain a satisfactory class of solutions. K

;

WHAT'S NEW FOR THE BELTRAMI EQUATION ?

141

Theorem 10.2. Suppose the distortion function K = K (z) for the Beltrami equation is such that e 2 L (B) for some positive p. Then K

p

the equation admits a unique principal solution h (19) h 2 z + W 1 (C ); Q(t) = t2 log,1 (e + t) ;Q

Moreover, every W 1 ( ) solution is factorisable. ;Q

loc

11. Distortion in the Subexponential Class We assume here that the Beltrami coecient is supported in the unit disk B. Theorem 11.1. There is a number p  1 such that every Beltrami equation whose distortion function has   K (z ) exp 1 + log K (z) 2 L (B) for p > p , admits a unique principal solution h 2 z + W 1 (C ) with Orlicz function Q(t) = t2 log,1 (e + t). Moreover we have  Modulus of Continuity; p

;Q

(20)

jh(a) , h(b)j2  log log(1C + K

j , j) 1

a

b

for all a; b 2 2B.  Inverse; The inverse map g = h,1 (w) has nite distortion K = K(w) and log K 2 L1 (C )  Factorization; each solution g 2 W 1 ( ) to the equation ;Q

loc

g = ( z ) g ; z

z

a:e:

admits a Stoilow factorisation (21) g (z ) =   h(z ) where  is holomorphic in h,1 ( ). In particular, all non-constant solutions in W 1 ( ) are open and discrete. ;Q

loc

12. Existence Theory. Various reductions show the important case to be when the Beltrami coecient  is compactly supported in the unit disk B. Then any solution is analytic outside the unit disk.

142

TADEUSZ IWANIEC AND GAVEN MARTIN

12.1. Results from Harmonic Analysis. The existence proof presented here exploits a number of substantial results in harmonic analysis. The arguments clearly illustrate the important r^ole that the higher integrability properties of the Jacobians have to play. The critical exponent p in Theorem 10.1 depends only on the constants in three inequalities which we now state. The rst is a direct consequence of [8]. Theorem 12.1. (Coifman, Lions, Meyer, Semmes) The Jacobian determinant J (x; ) of a mapping  2 W (C ) belongs to the Hardy space H (C ) and we have the estimate 0

1;2

1

kJ (x; )k

(22)

H 1 (C )

Z

C

1

C

jDj

2

Next we have from [9] Theorem 12.2. (Coifman, Rochberg) Let  be a Borel measure in C such that its Hardy{Littlewood maximal function M (x; ) is nite at a single point (and therefore at every point). Then log M (x; ) 2 BMO(C ) and its norm is bounded by an absolute constant, (23) k log M (x; )k  C : Finally we shall need the constant C which appears in the H -BMO duality theorem of Feerman, [12]. Theorem 12.3. (Feerman) For K 2 BMO(C ) and J 2 H (C ) we have 2

BM O

1

3

1

(24)

Z



K (x) J (x) dx  C kK k 3

BM O

kJ k

H1

Having these prerequisites we can reveal that the exponent in Theorem 10.1 is (25) p = 8C C C : 0

1

2

3

13. Sketch of Proof for Theorem 10.1 We again refer the reader to [14] for more details, but the basic ideas can be found here. Let K be the distortion function. Set

A =

Z

p

C



e

pK (z )



, e dz < 1 p

WHAT'S NEW FOR THE BELTRAMI EQUATION ?

143

We approximate  by smooth functions  via molli cation. We have K  K and the uniform bound Z

(26)

 pK (z )  C

e





, e dz  p



Z

 pK (z ) C

e



, ep dz = Ap

Put = to see e , e 2 L (C ). Next, the maximal function of e K z is nite everywhere, M (z; e K ) = e + M (z; e K , e )) (27) and this last term is a constant plus a function in L (C ). Now consider the BMO functions K (z) = 1 log M (z; e K ) (28) p

K (z )

2

2

( )

2

By Theorem 12.2, the BMO norm of this function does not depend on , (29) kK k  2C

p

 BMO

2

Moreover this function pointwise majorises the distortion function. We need a uniform L bound for K . Clearly 1 Z K (z) dz j2Bj B  Z 1 = log [M (z; e K )] 4 j2Bj B Z  41 log j21Bj [M (z; e K , e ) + e ] B   Z 2 C 1

K

 4 log 2e + j2Bj (e , e ) C Here we have used the L inequality for the maximal operator. This gives 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Z

(30)

jK (z)j dz  C log (1 + Ap) 2

2

B

4

2

where C is an absolute constant. Let us now return to the molli ed Beltrami equation, fz =  (z)fz (31) We look for a C {solution of (31) in the form fz (z) = e z ; fz (z) =  (z)e z (32) 4

1

( )

( )

144

TADEUSZ IWANIEC AND GAVEN MARTIN

where  2 W 1;p (C ; C ), for some p > 2, is compactly supported. The necessary and sucient condition for  is that (e )z = ( e )z or equivalently (33) z =  z + ( )z This equation is uniquely solved using the Beurling{Ahlfors transform, that is the singular integral operator de ned by Z 1 d ^ d (Sg)(z) = , 2i g((z)j,  )2 C Note that z = Sz and so equation (33) reduces to (34) (I ,  S )z = ( )z As kS k2 = 1 and as k k1 < 1, there is p > 2 such that k k1kS kp < 1 In this case the operator I ,  S has a continuous inverse. Thus (35) z = (I ,  S ),1 ( )z 2 Lp (C ) and also z = Sz 2 Lp (C ) (36) Note that z vanishes outside the support of  which is contained in B(0; 2). Also z = Sz = O(z,2) as z ! 1. Thus (z)  Cz asymptotically, for a suitable constant C . In fact Z 1 z ( ) (37)  (z ) = (T z )(z ) = 2i C  , z d ^ d where T is the complex Riesz Potential. Hence  is Holder continuous with exponent 1 , p2 , by the Sobolev Imbedding Theorem. Now the solution f  of equation (32) is unique up to a constant as fz = 0 outside 2B and as fz , 1 2 Lp (C ). That is, f  is a principal solution to the Beltrami equation (31). It is important to realise here that the Jacobian of f  is strictly positive, (38) J (z; f  ) = jfz j2 , jfz j2 = (1 , j j2 )e2 > 0 The Implicit Function Theorem tells us that f  is locally one-to-one. Another observation to make is that limz!1 f  (z) = 1. It is an elementary topological exercise to show that f  : C ! C is a global homeomorphism of C . It's inverse is C 1{smooth of course.

WHAT'S NEW FOR THE BELTRAMI EQUATION ?

145

We now digress for a second to outline the existence proof in the classical setting where K (z)  K < 1. As the sequence K is uniformly bounded we nd there is an exponent p = p(K ) > 2 such that kfz kp + kfz , 1kp  CK (39) where CK is a constant independent of  . Hence the Sobolev Imbedding Theorem yields the uniform bound (40) jf  (a) , f  (b)j  CK ja , bj1, p2 + ja , bj The same inequality holds for the inverse map and hence (41)

p j a , bj p, jf (a) , f (b)j  CK + ja , bj p,





2

2

2

We may assume that f  (0) = 0. As the p{norms of fz and fz , 1 are uniformly bounded, we may assume that each converges weakly in Lp(C ) after possibly passing to a subsequence. From the uniform continuity estimates and Ascoli's Theorem, we may further assume f  ! f locally uniformly in C . Obviously f satis es the same modulus of continuity estimates and is therefore a homeomorphism. Moreover, it follows that the weak limits of fz and fz , 1 must in fact be equal to fz and fz , 1 respectively. Hence f is a homeomorphism in the Sobolev class z + W 1;p(C ), that is fz and fz , 1 in Lp(C ). Finally observe that  !  pointwise almost everywhere, and hence in Lq (C ), where q is the Holder conjugate of p. The weak convergence of the derivatives shows that f is a solution to the Beltrami equation. Back to the more general setting. If we followed the above argument we nd the Lp bounds are useless as we cannot keep them uniform. We therefore seek an alternative route via a Sobolev-Orlicz class where uniform bounds might be available. We note the elementary inequality (42) (juj + jvj)2  2K (juj2 , jvj2) + 4K 2jv , wj2 ,1 juj and whenever u; v; w are complex numbers such that jwj  KK +1 K  1. We apply this inequality pointwise with u = z ; v = z ; w =  z and K = K (z) as de ned at (28), where (43)  (z ) = f  (z ) , z 2 W 1;2 (C ); K = K (z ) and use equations (31), (29) we can write (jz j + jz j)2  2K (jz j2 , jz j2) + 4(K )2j j2

146

TADEUSZ IWANIEC AND GAVEN MARTIN

and hence

jD (z)j  2K J (z;  ) + 4j K j

(44)

2

2

Next we integrate this and use Theorems 12.1 and 12.3 to obtain Z

jD j  2C kK kBM O kJ (z;  )kH + 4 

C

2



3

 4C Cp C 1

2

3

Z

1

jK j

2

B

2

jD j + 4C log (1 + Ap) 2

C

Z

2

4

where in the latter step we have used the uniform bounds at (29) and (30). It is clear point why we have chosen p = 8C C C at (25). R at this  The term C jD j in the right hand side can be absorbed in the left hand side. After doing this we obtain the uniform bounds in L 0

2

1

2

3

2

Z

(45)

jD j  8C log (1 + Ap) 2

C

4

which read as

kD kL

(46)



2 (C )

 C log

2

Z

5

B



e

pK

and in turn leaves us with the local estimate for the mapping f  (z) =  (z) + z, namely

kDf kL

(47)



2(

BR )  C5



R + log

Z

B

e

 pK

where BR = B(0; R). As f  is monotone (being a C homeomorphism) we can apply the modulus of continuity estimate of Theorem 5.2, 1

,R




pK Be   jf  (a) , f  (b)j  C R +1 log log 2 e + jaR,bj

(48)

6

for all a; b 2 BR . Now consider the inverse map to f  . Let us denote it by h = (f  ), : C ! C . As both f  and h are smooth dieomorphisms we 1

WHAT'S NEW FOR THE BELTRAMI EQUATION ?

nd

Z BR

j



Dh

( )j w

2

Z

=

dw

ZBR

=

(

K w; h

Z B

h ( R )

 

) (

J w; h

(

K z; f

( ( ) , 1) K z

C



Cp R

2

Z

+

B



)



)

147

dw

dz

+ j  (BR )j

dz

h

( ( ) , 1) K z

dz

In this last inequality we have put in the uniform bound j  (BR)j  . One interesting way to see this estimate (though perhaps not p the easiest) is via the Koebe distortion theorem. Anyway, we have h

C R

2

Z

(49)

B(0;R)

j



Dh

j  ( + 2

C

R

Z

2

B

K

)

and consequently we have the continuity estimate at (5.2) for  ,

j  ( ) ,  ( )j 

(50) For

h

f



x

h

y

2

this reads as

(51)

CR

log



e



R 2

h

BK 

+ jxR,yj R

j ( ) , ( )j  exp , j , Bj 2 B(0 ) and  1. The uniform f



a

f



b

CR

R

2

a

b

K



2

; whenever bounds, and the continuity estimates from above and below now enable us to pass to the limit. We nd  ! and  ! = , locally uniformly in  C and and  converging weakly in loc(C ). As in the classical setting this implies that is a homeomorphic solution to the Beltrami equation. Moreover z z , 1 2 (C ) and the same is true of the inverse function a; b

;R

R

f

Df

f

h

h

f

W

12

1

2 L

Dh

f

f ;f

2 L

References

[1] L.V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton 1966; Reprinted by Wadsworth Inc. Belmont, 1987. [2] K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173, (1994), 37{60. [3] K. Astala, T. Iwaniec, P. Koskela and G.J. Martin, Mappings with BMO bounded Distortion, Math. Annalen, 317, (2000), 703{726. [4] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63, (1977), 337-403.

148

TADEUSZ IWANIEC AND GAVEN MARTIN

[5] B. Bojarski, Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk. SSSR, 102, (1955), 661{664. [6] M.A. Brakalova and J.A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math., 76, (1998), 67{92. [7] R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certain Espaces Homogenes, Lecture Notes in Math., 242, Springer-Verlag, 1971. [8] R.R. Coifman, P.L Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72, (1993), 247{286. [9] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc., 79, (1980), 249{254. [10] R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103, (1978), 569{645. [11] G. David, Solutions de l'equation de Beltrami avec kk1 = 1, Ann. Acad. Sci. Fenn. Ser. AI Math., 13, (1988), 25{70. [12] C. Feerman, Characterisations of bounded mean oscillation, Bull. Amer. Math. Soc., 77, (1971), 587{588. [13] F.W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103, (1962), 353{393. [14] T. Iwaniec and G.J. Martin The Beltrami Equation, Memoirs of the Amer. Math. Soc., to appear. [15] T. Iwaniec and G.J. Martin Geometric Function Theory and Non-linear Analysis, to appear, Oxford University Press. [16] T. Iwaniec , P. Koskela, G.J. Martin and C. Sbordone Mappings of exponentially integrable distortion, to appear, [17] J. Manfredi, Weakly monotone functions, J. Geometric Analysis, 3, (1994), 393-402. [18] C.B. Morrey, On the solutions of quasi-linear elliptic partial dierential quations, (1938), 126{166. [19] S. Muller, A surprising higher integrability property of mappings with positive determinant, Bull.Amer. Math. Soc. 21 (1989), 245{248. [20] Rochberg and Weiss Analytic families of Banach spaces and some of their uses, Recent progress in Fourier analysis (El Escorial, 1983), 173{201, North-Holland Math. Stud., 111, North-Holland, Amsterdam-New York, 1985. [21] V. Ryazanov, U. Srebro and E. Yacubov, BMO{quasiregular mappings, To appear, J. D'Analyse Math. [22] E.M. Stein, Note on the class L log L, Studia Math., 32, (1969), 305{310. T. Iwaniec, Department of Mathematics, Syracuse University, Syracuse NY, USA

E-mail address :

[email protected]

G.J. Martin, Department of Mathematics, The University of Auckland, Auckland, NZ

E-mail address :

[email protected]

SOME SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS ASSOCIATED WITH LIE GROUPS PALLE E. T. JORGENSEN To Derek Robinson on the occasion of his 65th birthday

Abstract. In this note we survey results in recent research papers

on the use of Lie groups in the study of partial dierential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.

1. Introduction When the Hamiltonian of a quantum-mechanical system is related to a Lie algebra, it is often possible to use the representation structure of the Lie algebra to decompose the Hilbert space of the quantummechanical system into simpler (irreducible) pieces. For example, if a Hamiltonian commutes with the generators of a Lie algebra, the Hilbert space of the system can be decomposed into irreducibles of the Lie algebra, and the Lie algebra elements themselves can be used as elements in a set of commuting observables. We have aimed at making the present paper accessible to a wide audience of non-specialists, stressing the general ideas and motivating examples, as opposed to technical details. 2000 Mathematics Subject Classi cation. Primary 35B10; Secondary 22E25, 22E45, 31C25, 35B27, 35B45, 35C99, 35H10, 35H20, 35K10, 41A35, 43A65, 47F05, 53C30. Key words and phrases. approximating variable coecient partial dierential equation with constant coecients, t ! 1 asymptotics, boundary value problem, Gaussian estimates, heat equation, Hilbert space, homogenization, nilmanifold, parabolic, partial dierential equations, scaling and approximation of solution, spectrum, strati ed group. This research was partially supported by two grants from the U.S. National Science Foundation, and by the Centre for Mathematics and its Applications (CMA) at The Australian National University (ANU). This paper is an expanded version of a lecture given by the author at the National Research Symposium on Geometric Analysis and Applications at the ANU in June of 2000. 149

150

PALLE E. T. JORGENSEN

The class of such Hamiltonians is quite large: see [JoKl85] and [Jor88]. In this introduction we will review those Hamiltonians H whose interaction terms are polynomial in the position variables. Such Hamiltonians are directly and naturally related to nilpotent Lie algebras. The nilpotent case is studied in Section 2. The spectrum of H is obtained by decomposing the physical space on which the Hamiltonian H acts into irreducible representations of the underlying nilpotent group. Sometimes this decomposition is decisive, as is the case with a particle in a constant magnetic eld, where the decomposition leads to a harmonic-oscillator Hamiltonian. Sometimes the decomposition leads to a new Hamiltonian that requires further analysis, as is the case with a particle in a curved magnetic eld. The time evolution of the system is obtained by solving the heat equation of the underlying nilpotent Lie group. By writing the Hamiltonian as a quadratic sum of Lie-algebra elements and then using the representation of these Lie-algebra elements arising from the regular representation, it is possible to write e,tH as the convolution of a kernel (which is a solution of the heat equation) with a representation acting on the physical Hilbert space; see [Jor88]. The simplest case of this spectral picture is as follows: Consider a nonrelativistic spinless particle of mass m in an external magnetic eld B (x). The Hamiltonian for such a system is given by  2 1 e H = 2m p , c A ; (1.1) where p = hi r and A is the vector potential satisfying B = r  A. Consider the commutators i h e e pi , c ai ; pj , c aj = , hi ec "ijk bk ; h i @bj  h b ; pi , ec ai ; bj = hi @x (1.2) ij i i i h @b h e pi , c ai; bjk = i @xjk  hi bijk ; i ... ... ; where A = (a1 ; a2; a3), B = (b1 ; b2; b3 ), x = (x1 ; x2; x3 ). If B is a polynomial in x, eventually the derivatives of B will give zero, so that the set of commutators closes. The resulting Lie algebra formed by real linear combinations of the elements pi , ec ai; bi; bij ; : : : (1.3)

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS

151

is therefore a nilpotent Lie algebra, and the Hamiltonian is
qua, (1.1) e dratic in the rst three Lie algebra elements Xi := pi , c ai , i = 1; 2; 3, from the list (1.3). By general theory, e.g., [Rob91], this Lie algebra is the Lie algebra g of some Lie group G, which we may take to be simply connected. We show further in [JoKl85] and [Jor88] that there is a unitary representation U of G on L (R ) such that !  X e 2mH = dU pi , c ai : i If there is a constant of motion for the Lie-algebra elements pi , ec ai, then U is a direct integral over a corresponding spectral parameter R  . We then get H =  d H  where H has absolutely continuous spectrum, while each H  has purely discrete spectrum. If  ( )   ( ) 


is the spectrum of H  , then each  7! i ( ) is real analytic, and we get the following typical spectral picture. 2

3

3

2

=1

( )

( )

1

0

( )



 ( ) 2

 ( ) 1

 ( ) 0

 In this paper we will focus attention on a more restricted case wherein the coecients are periodic. As shown in Section 3, this case shares the spectral band structure with the polynomial-magnetic- eld case. We show that in the periodic case the regularity of the coecients may be relaxed, and in fact, our spectral-theoretic results will be valid when the operator has L1 -coecients. 2. Periodic operators We begin by recalling some elementary de nitions and facts about strati ed Lie groups from [FoSt82]. A real Lie algebra g is called

152

PALLE E. T. JORGENSEN

strati ed

if it has a vector-space decomposition

(2.1)

g=

r M

g(

k)

;

k =1

for some r, which we shall take nite here, all but a nite number of the subspaces g are nonzero,   (2.2) g ;g  g for all k; l 2 N , and g generates g as a Lie algebra. Thus a strati ed Lie algebra is automatically nilpotent, and if r is the largest integer such that g 6= 0, then g is said to be nilpotent of step r. A Lie group is de ned to be strati ed if it is connected and simply connected and its Lie algebra g is strati ed. Let G be a strati ed Lie group and exp: g ! G the exponential map. The Campbell{Baker{Hausdor formula establishes that exp (X ) exp (Y ) = exp (H (X; Y )) ; where H (X; Y ) = X + Y + [X; Y ] =2 + a nite linear combination of higher-order commutators in X and Y . Thus X; Y ! H (X; Y ) de nes a group multiplication law on the underlying vector space V of g which makes V a Lie group whose Lie algebra is g and the exponential map exp: g ! V is simply the identity. Then V with the group law is dieomorphic to G. Next let d denote the dimension of g and d the =  g and for each k choose a vector-space basis X  dimension of X ; : : : ; X k of g such that X ; : : : ; X = X ; : : : ; X r is a basis of g. If  ; : : : ;  is the dual basis for g , i.e., if  (X ) =  , de ne  =   exp, . Then  ; : : : ;  are a system of global coordinates for G, and the product rule on G becomes  (xy) =  (x) +  (y) + P (x; y) ; x; y 2 G; where P (x; y) is a nite sum of monomials in  (x),  (y) for i < k with degree between 2 and m. It follows that both left and right Haar measure on G can be identi ed with Lebesgue measure d


d . If X denotes one of the (abstract) Lie generators, we denote by A the corresponding right-invariant vector eld on G, i.e., A on a test function on G is given by A = dL (X ), or more precisely,   d (exp (,tX ) g) j ; g 2 G; A (g) = dt (2.3) and similarly A = dR (X ) given by   d (g exp (tX )) j : A (g) = dt (2.4) (k )

(k )

(l)

(k +l)

(1)

(r )

(k )

k

(k ) 1

(k )

1

d

1

k

(k )

(k )

(1) 1

d

d

k

1

k

1

k

(r )

d

l

k;l

d

k

k

k

k

i

i

1

d

i

i

i

(l)

i

i

(l)

i

i

t=0

(r )

i

i

(r ) i

i

t=0

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS

153

Since we can pass from left to right with the adjoint representation, the formulas may be written in one alone, and we will work with Ail , and denote it simply Ai . If 1  j  d we will need the functions yj on G de ned by ( )

1

yj exp

(2.5)

d X

k Xk

!!

= j :

k =1

These functions satisfy the following system of dierential equations: (2.6) ,Ail yj = Air yj = i;j : It follows by the standard ODE existence theorem that the functions yi on G are determined uniquely by (2.6) and the \initial" conditions yi (e) = 0. Also note that (2.6) is consistent only for the dierential equations de ned from a sub-basis A ; : : : ; Ad1 , and that they would be overdetermined had we instead used a basis: hence the distinction between subelliptic and elliptic. In addition, we have given a discrete subgroup , in G such that M = G=, is compact. It is well-known that it then has a unique (up to normalization) [Jor88, Rob91] invariant measure . The corresponding Hilbert space is L (M; ), and the invariant operators on G pass naturally to invariant operators on M ; see [BBJR95]. Let X ; : : : ; Xd1 be the generating Lie-algebra elements. Then the corresponding invariant vector elds on G will be denoted A ; : : : ; Ad1 , and those on M will be denoted B ; : : : ; Bd1 . Functions ci;j 2 L1 (G) are given, and we form the quadratic form ( )

( )

1

2

1

1

1

(2.7)

h (f ) =

d1 X i;j =1

hAif j ci;j Aj f i :

If further (2.8) ci;j (g ) = ci;j (g) for g 2 G; 2 ,; then we have a corresponding form hM on M = G=,. Introducing ,
c"i;j (x) = ci;j ", x ; " > 0; (2.9) we get for each " a periodic problem corresponding to the period lattice ",. To speak about ", for " 2 R , we must have an action of R on G which generalizes the familiar one " : (x ; : : : ; xd) 7,! ("x ; : : : ; "xd ) d of R . It turns out that this can only be done if G is strati ed, and so in particular nilpotent; see [FoSt82], [Jor88]. In that case it is possible to 1

+

1

+

1

154

PALLE E. T. JORGENSEN

construct a group of automorphisms f"g"2R+ of G which is determined by the dierentiated action d" on the Lie algebra g. If g is speci ed as in (2.1){(2.2), then ,
(2.10) d" X = "X ; X 2 g ; " 2 R : Let H , respectively H", be the selfadjoint operators associated to the period lattices , and ", (see [BBJR95] or [Rob91]), and let St = e,tH , St" = e,tH" . We now turn to the homogenization analysis of the limit " ! 0 which leads to our comparison of the variable-coecient case to the constant-coecient one. It should bePstressed that in the Lie case, even the \constant-coecient" operator i;j Ai c^i;j Aj is not really constantcoecient, as the vector elds Ai are variable-coecient. Take even the simplest example where G is the three-dimensional Heisenberg group of upper triangular matrices of the form 01 x z1 g = @0 1 yA ; x; y; z 2 R: (2.11) 0 0 1 In this case, dim g = 2, and dim g = 1, with g spanned by the central element in the Lie algebra. Dierentiating matrix multiplication (2.11) on the left as in (2.3), we get the following three identities: (1)

(1)

(1)

(1)

(1)

(2)

+

(2)

@ + y @ = ,dL (X ) ; A = @x @z @ A = @y = ,dL (X ) ; @ = ,dL (X ) ; A = @z 1

1

2

2

3

3

where the rst vector eld is of course variable coecients. We will use standard tools [ZKO94] (see also [Dau92], [Tho73], [Wil78]) on homogenization. Theorem 2.1. [BBJR95] Suppose the system ci;j 2 L1 is given and assumed strongly elliptic. Then there is a C -semigroup S^t on L (G; dx) with constant coecients, where dx is left Haar measure, such that

 " 

lim

St , S^t f = 0 "! for all f 2 L (G; dx) and t > 0. The constant coecients of the limit operator c^i;j may be determined as follows: We show in [BBJR95] that if (2.12) ci;j (g) := h (gi , yi; gj , yj ) 0

0

2

2

2

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS

155

and if C (g) is the corresponding quadratic form, then the problem (2.13) inf C (g) =: C^ g

has a unique solution, i.e., the in mum is attained at f1 ; : : : ; fd1 such that ^ (2.14) C (f ) = C: The order relation which is used in the in mum consideration (2.13) is the usual order on hermitian matrices: For every g, the matrix C (g) := (ci;j (g))di;j1 =1 is hermitian, and the matrix inequality C (g)  C^ may thus be spelled out as follows: X X zi ci;j (g) zj  zic^i;j zj for all z1; : : : ; zd1 2 C : i;j

i;j

Solvability of this variational problem is part of the conclusion of our analysis in [BBJR95], i.e., the existence of the minimizing functions f1; : : : ; fd1 . Then the coecients of the homogenized operator can also be computed with the aid of the coordinates yi, i = 1; : : : ; d1, introduced in (2.5) and (2.6). One has the representation (2.15)

c^i;j =

Z

Y

dy

d1 X k;l=1

(Ak (fi (y) , yi)) ck;l (y) (Al (fj (y) , yj ))

= hY (fi , yi; fj , yj ) ; where h denotes the sesquilinear form associated with H , and the subscript Y refers to the region of integration. Speci cally, Y is a fundamental domain for the given lattice , in G. For example, we may take Y to be de ned by \ (2.16) x 2 G ; jxj  x ,1 ; Y=

2,

and j
j de ned relative to a geodesic distance d, jxj := d (x; e), x 2 G. Then S (i) 2, Y = G, and (ii) meas (Y 1 \ Y 2) = 0 whenever 1 6= 2 in ,. (These are the axioms for fundamental domains of given lattices, but we stress that (2.16) is just one choice in a vast variety of possible choices.) The simplest case of the construction is G = R, and it was rst considered in [Dav93, Dav97] by Brian Davies. This is the simplest

156

PALLE E. T. JORGENSEN

possible heat equation, and we then have the conductivity represented by a periodic function c, say c (x + p ) = c (x) ; x 2 R ; where p is the period. Then H = , dxd c (x) dxd , and it can be checked that  Z p0 , 1 dx c^ = p c (x) : Theorem 2.2. [BBJR95] Adopt the assumptions of Theorem 2.1. Then D= ^ lim t ess sup Kt (x ; y ) , Kt (x ; y ) = 0 t!1 2 2 0

0

1

0

0

2

jxj +jyj at

for each a > 0 where jxj = dc (x ; e), and where

dc (x ; y) = sup

(

(x) , (y) ; 2 Cc1 (G) ; d1 X i;j =1

ci;j (Ai ) (Aj )  1 pointwise

)

and Ai refers to the Lie action of the vector eld Ai on from (2.3). It is our aim here only to sketch the ideas, and the reader is referred to our papers for details, but we stress that the proof is based on homogenization, see, e.g., [BLP78], [ZKON79], [Koz80], and [AvLi91] The number D is the homogeneous degree de ned from the given grading, or strati cation, g(i) of the nilpotent Lie algebra g. As spelled out in [Jor88] and [FoSt82], there are numbers i depending on the Lie-structure coecients such that X D = i dim g(i) : i

To be speci c, the numbers i are determined in such a way that we get a group of scaling automorphisms f"g"2R+ of g, and therefore on G, and it is this group which is fundamental in the homogenization analysis. Speci cally, extending (2.10), " : g ! g is de ned by ,
" X i = " X i ; X i 2 g i ; (2.17) and then extended to g by linearity via (2.1), in such a way that (2.18) " ([X; Y ]) = [" (X ) ; " (Y )] ; X; Y 2 g; " 2 R : Hence if (2.2) holds, then it follows from (2.17) and (2.18) that i = i for i = 1; 2; : : : . In the case of the Heisenberg Lie algebra g, we have ( )

i

( )

( )

( )

+

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS

157

[X; Y ] = Z as the relation on the basis elements; Z is central. Then g = span (X; Y ), g = R Z ,  = 1,  = 2, so D = 4. Let Kt and K^ t be the respective integral kernels for the semigroups St and S^t , and set (1)

(2)

1

jjK jjp = xess 2G

and

2

Z

G

j

j

dy K (x; y )

p

1=p

jjK jj1 = x;yess2G jK (x; y)j :

Then

Theorem 2.3. [BBJR95] Adopt the assumptions of Theorem 2.1. Then ^ t = 0: ^ t = 0; lim lim tD= Kt , K Kt , K t!1 t!1 1 2

1

3. G = R d The case G = R d was considered in [BJR99], where we further showed that the limit St" ! S^t then holds also in the spectral sense. In that case, we scale by " = 1=n, n ! 1, and then identify the limit operator as having absolutely continuous spectral type, and we prove spectral asymptotics. (A general and classical reference for periodic operators is [Eas73].) Starting with an equation which is invariant under the Zd-translations, we then use the Zak transform [Dau92] to write St = e,tH as a direct integral over Td (= R d =Zd), viz., (3.1)

St

=

Z 

Td

(z )

St ;

and we establish continuity of z 7! St z in the strong topology [BJR99, Lemma 2.2]. Pick a positive C 1-function  on R d of integral one, and set Z n d dy  (ny ) ci;j (x , y ) ; ci;j (x) = n ( )

( )

Rd

and form the corresponding C -semigroup n n St = e,tH ; where H n is de ned from ci;jn . We then show in [BJR99] that St n approximates St , not only in the strong topology, but also in a spectraltheoretic sense. Using this, we establish the following connection be
, i z id , tH , tH z in (3.1). Setting z = e ; : : : ; e , tween St = e and St = e we get 0

( )

( )

( )

( )

( )

( )

( )

1

158

PALLE E. T. JORGENSEN

Theorem 3.1. If  (z) denotes the eigenvalues of H then (3.2) lim !1 N



n

N 2 n (w ) ; w N

= z; n = 0; 1; : : : nD



z

E

o

= (n , ) j C^ (n , ) ; n 2 Z ; where the limit is in the sense of pointwise convergence of the ordered sets, and where C^ = (^c ) is the constant-coecient homogenized case. The rate of convergence of the eigenvalues in (3.2) can be estimated further by a trace norm estimate. We refer the reader to [BJR99] for details of proof, but the arguments in [BJR99] are based in part on the references [Aus96], [DaTr82], [Eas73], and [ZKON79]. In addition, we mention the papers [Aus96], [AMT98], and [TERo99], which contain results which are related, but with a dierent focus. Finally, we mention that our result from [BJR99], Theorem 3.1, has since been extended in several other directions: see, e.g., [Sob99] and [She00]. Acknowledgements. We are grateful to Brian Treadway for excellent typesetting and graphics production, and to the participants in the National Research Symposium at The Australian National University for fruitful discussions, especially A.F.M. ter Elst. d

i;j

[Aus96] [AMT98] [AvLi91] [BBJR95] [BLP78] [BJR99] [DaTr82] [Dau92] [Dav93]

References

P. Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc. (2) 54 (1996), 284{296. P. Auscher, A. McIntosh, and P. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1998), 22{73. M. Avellaneda and F.-H. Lin, Lp bounds on singular integrals in homogenization, Comm. Pure Appl. Math. 44 (1991), 897{910. C.J.K. Batty, O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson, Asymptotics of periodic subelliptic operators, J. Geom. Anal. 5 (1995), 427{443. A. Bensoussan, J.-L. Lions, and G.C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. O. Bratteli, P.E.T. Jorgensen, and D.W. Robinson, Spectral asymptotics of periodic elliptic operators, Math. Z. 232 (1999), 621{650. B.E.J. Dahlberg and E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helv. 57 (1982), 130{134. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992. E.B. Davies, Heat kernels in one dimension, Quart. J. Math. Oxford Ser. (2) 44 (1993), 283{299.

PARTIAL DIFFERENTIAL EQUATIONS AND LIE GROUPS [Dav97] [Eas73] [FoSt82] [Jor88] [JoKl85] [Koz80] [Rob91] [She00] [Sob99] [TERo99] [Tho73] [Wil78] [ZKO94] [ZKON79]

159

E.B. Davies, Limits on Lp regularity of self-adjoint elliptic operators, J. Dierential Equations 135 (1997), 83{102. M.S.P. Eastham, The Spectral Theory of Periodic Dierential Equations, Scottish Academic Press, Edinburgh, Chatto & Windus, London, 1973. G.B. Folland and E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, 1982. P.E.T. Jorgensen, Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics, North-Holland Mathematics Studies, vol. 147, Notas de Matematica, vol. 120, North-Holland, Amsterdam{New York, 1988. P.E.T. Jorgensen and W.H. Klink, Quantum mechanics and nilpotent groups, I: The curved magnetic eld, Publ. Res. Inst. Math. Sci. 21 (1985), 969{999. S.M. Kozlov, Asymptotics of fundamental solutions of second-order divergence dierential equations, Mat. Sb. (N.S.) 113(155) (1980), no. 2(10), 302{323, 351 (Russian), English translation: Math. USSRSb. 41 (1982), no. 2, 249{267. D.W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Z. Shen, The periodic Schrodinger operators with potentials in the Morrey{Campanato class, preprint, University of Kentucky, 2000. A.V. Sobolev, Absolute continuity of the periodic magnetic Schrodinger operator, Invent. Math. 137 (1999), 85{112. A.F.M. ter Elst and D.W. Robinson, Second-order subelliptic operators on Lie groups, I: Complex uniformly continuous principal coecients, Acta Appl. Math. 59 (1999), 299{331. L.E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335{343. C.H. Wilcox, Theory of Bloch waves, J. Analyse Math. 33 (1978), 146{ 167. V.V. Z ikov, S.M. Kozlov, and O.A. Olenik, Homogenization of Dierential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, translated by G.A. Yosi an from the Russian Usrednenie differencial~nyh operatorov, \Nauka", Moscow, 1993. V.V. Z ikov, S.M. Kozlov, O.A. Olenik, and H.T. Ngoan, Averaging and G-convergence of dierential operators, Uspekhi Mat. Nauk 34 (1979), no. 5(209), 65{133, 256, Russian Math. Surveys 34 (1979), no. 5, 69{ 148.

Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, U.S.A.

E-mail address : [email protected] URL: http://www.math.uiowa.edu/~jorgen/

PRINCIPAL SERIES AND WAVELETS CHRISTOPHER MEANEY

Abstract. Recently Antoine and Vandergheynst [1, 2] have pro-

duced continuous wavelet transforms on the -sphere based on a principal series representation of ( 1). We present some of their calculations in a more general setting, from the point of view of Fourier analysis on compact groups and spherical function expansions. n

S O n;

1. Coherent States We begin with Antoine and Vandergheynst's de nition of a coherent state, as presented in [1, 2]. Here G is a locally compact group.  Suppose that X is a homogeneous space of G, X = G=H , equipped with a G-invariant measure.  Let (U; L2(Y )) be a unitary representation of G on some Lebesgue space L2 (Y ).  Assume there is a Borel cross section  : X ,! G;  (x) H = x; 8x 2 X:  Say that  2 L2 (Y ) is admissible mod(H; ) when Z

X

jhU ( (x)) j'ij2 dx < 1; 8' 2 L2 (Y ) :

 The orbit of an admissible vector  under  (X ), fU ( (x))  : x 2 X g

is called a coherent state. Note that there are other variations on the theme of \restricted square integrability", such as the case described in [3]. 1991 Mathematics Subject Classi cation. 43A90,22E46,43A75,42C40. Key words and phrases. Semisimple Lie group, coherent state, continuous wavelet transforms, principal series, Plancherel formula, admissible vectors. This is the content of my lecture at the National Research Symposium on Geometric Analysis and Applications in Canberra, June 2000. In the past year my research was partially supported by the ARC Small Research Grants Scheme. 160

PRINCIPAL SERIES AND WAVELETS

161

2. Frames Suppose now that  is an admissible vector in L (Y ). De ne a linear operator 2

A; : L (Y ) ,! L (Y ) 2

by

hA; ' j' i = 1

2

Z X

2

h' jU ( (x)) i hU ( (x)) j' i dx; 8' ; ' 2 L (Y ): 1

2

1

2

2

When this has a bounded inverse, say that the coherent state is a frame. When the orbit of  under  (X ) is a frame of L (Y ) there is the continuous wavelet transform, 2

W : L (Y ) ,! L (X ) 2

2

de ned by

W ' (x) = h'jU ( (x)) i ; 8' 2 L (Y ): 2

This operator is one-to-one and its range H is complete with respect to the inner-product:

hW 'jW iH = W 'jW A,; 1



L2 ( X )

;

; ' 2 L (Y ): 2

Hence there is a unitary isomorphism W : L (Y ) ,! H . 2

3. The setting For the calculations which we will describe here, the ingredients are:  G is a noncompact connected semisimple Lie group with nite centre and Cartan involution .  K is the corresponding maximal compact subgroup.  G = KAN is an Iwasawa decomposition.  M is the centralizer of A in K .  X = G=N .  Y = K=M .  U is a certain principal series action of G on L (K=M ), to be de ned below.  Assume that (K; M ) is a Gel'fand pair. See Knapp's book for details [5, page 119]. 2

162

CHRISTOPHER MEANEY

There are for which

Iwasawa

4. Decompositions projections K : G ! K , A : G ! A, N : G ! N ,

g = K(g)A(g)N(g); 8g 2 G: The Haar measure on G is given in terms of that of K and right Haar measure of AN , [5, page 139] with dg = dk dr (an): The measure on K is normalized so that Z

K

dk = 1:

There is a mapping log : A ! a with For each  2 a let

8a 2 A:

exp(log(a)) = a;

a = e

a

(log( ))

8a 2 A:

;

5. Invariant Integration There is the special functional  2 a determined by the structure of the group G. For f 2 Cc(G) the integral formula for Haar measure on G is Z

G

f (x) dx =

Z Z Z

f (kan) a  dndadk: 2

K A N

See [6, Prop. 7.6.4] for details. We can use KA to parametrize G=N and the G-invariant integral on G=N is given by Z

G=N

for F be

F (y)dy =

Z Z

F (kaN )a  dadk 2

K A

2 Cc(G=N ). Hence, we take the Borel section  : G=N ! G to (kaN ) = ka; 8a 2 A; k 2 K:

PRINCIPAL SERIES AND WAVELETS

163

6. Induced Representations Consider the space of continuous covariant functions: 9 8 f : G ! C continuous = < f (gman) = a,f (g); ; : I(G) = :f : 8g 2 G; m 2 M; a 2 A; n 2 N Left translation by elements of G preserves the property of covariance: ,
(U (g)f ) (x) = f g,1x ; 8g; x 2 G; f 2 I(G):

U (g) : I(G) ,! I(G); 8g 2 G: For a covariant function f 2 I(G), f (x) = f (K(x)A(x)N(x)) = A(x), f (K(x)); 8x 2 G: Equip I(G) with the inner product Z

hf1jf2i = and norm

kf k =

The completion of I(G) is

K

Z

f1(k)f2 (k) dk

jf (k)j dk 2

K

1=2

:

HU = L2 (K=M ):

The action of G on HU is an example of a principal series representation, see section 8.3 of Wallach's book [6]. For our purposes, the essential fact is that U jK is the regular representation of K on a subspace of L2(K ). If f 2 L2(K=M ), extend it to be an element of HU by assigning f (kan) = a,f (k): Notice that if f 2 L2 (K=M ), U (g)f (k) = A(g,1k),f (K(g,1)k); k 2 K; g 2 G: For each g 2 G the action of U (g) extends to a continuous linear operator on HU . It is a unitary representation: Z

hU (g)f1jU (g)f2i = (U (g)f1)(k)(U (g)f2)(k) dk =

K

Z

Lemma 1.

K

A(g ,1k ),2 f1 (K(g ,1 k))f2 (K(g ,1k)) dk: = hf1 jf2 i

(U; HU ) is unitary. When restricted to K , it is the action of K by left translation on L2 (K=M ). The representation

164

CHRISTOPHER MEANEY

7. Fourier analysis on the compact group K We review some basic facts about analysis on compact groups. Let K be the dual object of K , consisting of a maximal set of inequivalent irreducible unitary representations ( ; V ) of K . For each integrable function f on K there is the Fourier series: f (x) = d f   (x):

b

X

Z where the Fourier coecient is Z b( ) = ( ) (

b

2K

Convolution with a character is f   (x) = f (y ) tr( (y ,1) (x)) dy = tr(f ( ) (x)) K

f

K

f x x,1 ) dx =

b

Z

K

f (x) (x) dx:

The Fourier coecients are linear transformations f ( ) 2 HomC (V ; V ): Fourier coecients of convolutions are products of Fourier coecients:

b Z Z ) ( )= ( )( Z Z = ( )( = b( ) b( )

(f  g ^

K

K

K

K

f x g x,1 y ) (y ,1) dxdy

f x g x,1 y ) (y ,1xx,1 ) dxdy

g f :

De ne left translation on K by ,1 8x; y 2 K; x f (y ) = f (x y ); and the composition with inversion f _ (x) = f (x,1 ); 8x 2 K: Fourier coecients of left translates satisfy (x

f )^ ( ) =

Z

K

b

f (x,1 y ) (y ,1xx,1 ) dy = f ( ) (x,1 )

Fourier coecients of adjoints satisfy (g_)^( ) = g( ): The L2 (K ) inner product can be viewed as a convolution:

Z

K

f (x)g (x) dx

Z =

b

K

f (x)g_ (x,1 ) dx = f  g _ (1):

PRINCIPAL SERIES AND WAVELETS

165

For f; g 2 L2 (K ), the Fourier series of their convolution is absolutely convergent, see [4], X f  g(x) = d f  g   (x)

f and g in L2 (K ): Z K

b

2K

f  g(x) =

f (x)g(x) dx =

kf k22 =

X b

2K

X

b

2K

X

b

2K





d tr gb( )fb( ) (x) ; d tr(fb( )bg( ));

2

d

fb( )

 : 2

In particular, for each 2 Kb , kfb( )k22 = d kf   k22: See Appendix D of Hewitt and Ross [4] for details about the norms k
kp ; 1  p  1: If h 2 L1(K ) then f 7! f  h; L2(K ) ,! L2 (K ); is a bounded linear operator which commutes with left translation. Similarly, f 7! h  f; L2 (K ) ,! L2(K ); is a bounded linear operator which commutes with right translation. The norm of both of these operators is

sup

bh( )

 : b

2K

1

8. Homogeneous Spaces Now we return to dealing with functions on K=M , which we identify with right-M -invariant functions on K . For each 2 Kb , let V M = fv 2 V : (m)v = v; 8m 2 M g and P : V ,! V M , the orthogonal projection on to this subspace. Let  be the normalized Haar measure on M . Its Fourier coecients are b b( ) = P ; 8 2 K:

166

CHRISTOPHER MEANEY

If f 2 L1(K=M ) then b f = f  ; =) fb( ) = P fb( ); 8 2 K: We are restricting our attention to the case where (K; M ) is a Gel'fand pair, which means that ,
b dim V M  1; 8 2 K: Lemma 2. If (K; M ) is a Gel'fand pair and f 2 L1 (K=M ), then for all 2 Kb , rank(fb( ))  1 and (V M )?  ker(fb( ) ): Lemma 3. If (K; M ) is a Gel'fand pair and f 2 L1 (K=M ), then for all 2 Kb , fb( )fb( ) = kfb( )k22 P : Lemma 4. If (K; M ) is a Gel'fand pair and f 2 L1 (K=M ), then for all 2 Kb , kfb( )kp = kfb( )k2 ; 1  p  1: Lemma 5. If (K; M ) is a Gel'fand pair and h 2 L1 (K=M ), then the norm of the operator f 7! f  h; L2 (K ) ,! L2 (K=M ); is n o np o b b b sup kh( )k2 : 2 K = sup d kh   k2 : 2 K : ,




In this lemma, if dim V M = 0 then bh( ) =,0 and
so we need only take the supremum over those for which dim V M = 1. 9. Admissible Vectors In [2] the unitary representation (U; HU ) of G is said to be squareintegrable modulo N if there is a non-zero vector  for which Z Z

K A

jhU (ka)j ij2 a2 dadk < 1

for all  2 HU . Such an  is called admissible. Notice that this can be rearranged to say Z Z



 U (a) U (k ,1 ) 2 a2 dadk
, n4 and more precisely (11) C_1(X ) ,! D(
s) ,! C ,1(X ) are dense inclusions for , n4 < s < n4 : The limits of this range correspond to the occurrence of formal solutions of
u = 0:

Wave group

2. The Cauchy problem for the wave equation (Dt2 ,
)u = 0; on R  X  (12) ujt=0 = u0; Dt ujt=0 = u1 has a unique solution u 2 C 0(R ; D(
1=2 )) \ C 1(R ; L2g (x)) 8 (u0; u1) 2 E = E1 = D(
1=2 )  L2g (X ) : These ` nite energy solutions' are the main object of study here. More s ,1 ,1 2 generally, with the equation interpreted in C (R ; D(
)) the Cauchy problem has a unique solution u 2 C 0(R ; D(
s2 )) \ C 1(R ; D(
s2 , 12 )) (13) 8 (u0; u1) 2 Es = D(
2s )  D(
2s , 21 ) : The regularity hypothesis on the solution can be weakened to u 2 L2loc (R; D(
s2 )) \ Hloc1 (R ; D(
s2 , 21 )) without changing the unique solvability. Notice that these calculations are consistent under decrease of s: Furthermore, partial hypoellipticity in t shows that the solution to (12) satis es (14) u 2 Hloc,k (R ; D(
s2 + k2 )) 8 k 2 R : An admissible solution to the wave equation is one that satis es u 2 Hlocp (R ; D(
q2 )) for some q; p 2 R (15) ,p,2 (R ; D(
q2 ,1 )). Such a solution automatically with (4) holding in Hloc satis es (14) for some s: These statements can be reinterpreted in terms of the wave group     u ( t ) u 0 (16) U (t) : u 7! D u(t) ; U (t) : Es ! Es 8 s: t 1

176

RICHARD B. MELROSE AND JARED WUNSCH

3. Hormander's theorem Let M be a manifold without boundary. The wave front set of a distribution u 2 C ,1(M ) is a closed subset of the cosphere bundle WF(u)  S M: It may be de ned by decay properties of the localized Fourier transform, or the FBI (Fourier{Bros{Iagonitzer) transform, or by testing with pseudodierential operators. The projection (WF(u))  M is exactly the C 1 singular support, the complement of the largest open subset of M to which u restricts to be C 1 . A re ned notion of wavefront set is the Sobolev-based wavefront set, denoted WFs; this is a closed subset of S M , where now the projection is the complement of the largest open subset of M to which u restricts to be H s. If u satis es a linear dierential equation, P u = 0; then WF(u)  (P )  S M when (P ) is the characteristic variety of P , the set on which its (homogeneous) principal symbol, p; vanishes. If p is real then the symplectic structure on T M , or the contact structure on S M; de nes a `bicharacteristic' direction eld VP on S  M , tangent to (P ): The integral curves of VP are called bicharacteristics; those lying in (P ) are called null bicharacteristics. Theorem 1 (Hormander). Let P be a (pseudo)-dierential operator with real principal symbol. If P u = 0 then W F (u)  (P ) is a union of maximally extended null bicharacteristics. The same result also holds with WF replaced by WFs for any s: In our case, M = R  X  so T M = T R  T X  . The principal symbol of the d'Alembertian is  2 , j
j2g , where  is the dual variable to t and j
jg is the (dual) metric on T X . Then (P ) = + (P ) [ ,(P )  S M when I (P )  = R  S X  are the disjoint parts of (P ) in  > 0 and  < 0. In this representation of (P ) the null bicharacteristics are geodesics on X  , lifted canonically to S X , with t as ane parameter. Thus, for the wave equation over X  , Hormander's theorem does indeed reduce to the informal propagation statement described above. Combined with standard results relating the singularities of the solution to singularities of the initial data, Hormander's theorem applied to the wave equation on a conic manifold yields complete information on

SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES

177

the behavior of singularities except along bicharacteristics lying above geodesics which hit the boundary. 4. Diffractive theorem On parametrized geodesic segments with an end point on the boundary, the de ning function x is either strictly increasing or strictly decreasing near the boundary. For each sign of  and for each t 2 R the bicharacteristics covering such geodesics which hit the boundary at t = t and along which t is increasing (resp. decreasing) as x decreases, form a smooth submanifold of T (ft < tg X ) (resp. T (ft > tg X )). We denote these `radial' surfaces (near @X ) by R;I (t) and R;O (t)  (P ) where  is the sign of  and I; O refers to whether these are `incoming' or `outgoing' and hence, equivalently, whether they lie in t < t or t > t: Theorem 2 (Diractive theorem). If u is an admissible solution to (4) then for any t 2 R , s 2 R ,  = , R;I (t) \ WFs (u) = ; ) R;O (t) \ WFs (u) = ;: Here, WFs(u) is the wave front set computed relative to the Sobolev space H s; locally in the interior. This is a precise form of the diractive result described informally above. Notice that the singularities for dierent signs of  are completely decoupled. This does not, however, represent any re nement in terms of propagation along the underlying geometric rays, since all geodesics are covered by bicharacteristics with  xed and of either sign. The proof of this result is discussed brie y below in x7. 5. Geometric theorem Consider a geodesic on X which hits the boundary at a point p 2 @X: An open set of perturbations of the geodesic, meaning geodesics starting near some interior point on the geodesic and with initial tangent close to the tangent to the geodesic, will miss the boundary. A limit of such curves as the perturbation vanishes consists of three segments. The rst is the incoming geodesic segment. The second is a geodesic segment in the boundary, of length . The third is the outgoing geodesic from the end point of the boundary segment, which is therefore a point in G(p) as de ned in (5) (see Figure 2). Thus it is reasonable to suppose that, amongst the outgoing bicharacteristics leaving the boundary at time t; those with initial points in G(p) will be more

178

RICHARD B. MELROSE AND JARED WUNSCH

A sequence of geodesics nearly missing the boundary, and the three segments which they approach. Figure 2.

closely related to an incoming bicharacteristic with end point p arriving at time t: We call these the geometrically-related bicharacteristics (or geodesics). For instance, if there are incoming singularities on a single ray the singularities on the `non-geometrically-related' outgoing bicharacteristics might be expected to be weaker than the incoming singularity. However, this is not in general the case. To obtain such a geometric re nement of the diraction result we need to impose an extra `nonfocusing' assumption. Theorem 3 (Geometric theorem). Let u be an admissible solution to (4) and let  = . Suppose that R;I (t) \ WFs u = ; near @X . Suppose additionally that for some k and 0 < ` < n2 (17) WFs+`(1 +
@X ),k u \ R;I (t) = ;: For any 0 < r < ` , 1=2; if no incoming bicharacteristic hitting the boundary at time t at a point in G(p) with sgn  =  is in WFs+r u, then the outgoing bicharacteristic with initial point p 2 @X and sgn  =  is not in WFs+r, u for any  > 0. If in addition to (17) we have (18) WFs+`(xDx + (t , t)Dt)(1 +
@X ),k u \ R;I (t) = ;: then the same conclusion follows for all 0 < r < l. Thus the additional assumption (17) allows regularity on outgoing rays to be deduced from regularity in the incoming geometricallyrelated rays up to the corresponding level above `background' regularity.

SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES

179

As already noted, this result may be applied to the fundamental solution with initial point near the boundary. If the initial pole of the fundamental solution is suciently close to the boundary then there is a unique short geodesic segment from it to the boundary, arriving at a point p: If t is the length of the segment then, provided t is small enough, (17) and (18) hold with s < , n2 + 1 for any ` < n,1 2 . It  follows that on R;O (t); the outgoing set, the fundamental solution is in H 1=2,, for all  > 0, microlocally near the non-geometrically related rays, those with end point not in G(p), whereas the general regularity is H , n2 +1, for all  > 0. This is a gain of `nearly' n,1 2 derivatives over the background regularity. In this way we extend part of the result of Cheeger and Taylor [1, 2] in the product case (9) to the general conic case. Inspection of the fundamental solution constructed in [1] reveals the `nearly' n,1 2 dierence in smoothness between geometric and non-geometric rays to be sharp. 6. Spherical conormal waves Around a given point q in a compact Riemann manifold there are `spherical' conormal waves which are singular only on the spherical surfaces r = t, for small t of both signs. These just correspond to conormal data at t = 0 at the ( ctive) cone point q. An important example is the fundamental solution, in which case the result follows from Hadamard's construction. In the more general case of a conic manifold with boundary there are similar contracting, and then expanding, conormal waves. Theorem 4. If u is an admissible solution near @X and t = 0 which is conormal to t = ,x for t < 0 then it is conormal to t = x, near the boundary, for small t > 0. These conormal solutions to the wave equation in the general conic case are at the opposite extreme to those considered in the Geometric Theorem above. Namely, they are already smooth in the tangential variables, so no tangential smoothing in the sense of (6) is possible. Further analysis of the structure of these waves shows that the principal symbols undergo a transition at x = 0, the boundary, given by the scattering matrix for the model cone with the same boundary metric. Since this scattering matrix should have full support in general, this provides counterexamples to any extension of the geometric theorem in which the tangential smoothing condition is dropped.

180

RICHARD B. MELROSE AND JARED WUNSCH

7. Methods The basic method we use is microlocal, but non-constructive. It is a direct extension of one of the proofs by Hormander of the interior propagation theorem. This `positive' commutator method is itself a microlocalization of the energy method for hyperbolic equations. In it a `test' pseudodierential operator, A, is applied to the equation and the essential positivity of the symbol of the commutator 1 [P; A] gives a local regularity estimate on the solution. To extend this method to cover behavior of solutions near the boundary we replace the ordinary notions of wavefront set, pseudodierential operators and microlocalization with versions appropriately adapted to the geometry. When considering the Laplacian itself on the manifold with boundary with conic metric, the appropriate notion is that of a weighted b-pseudodierential operator (see [13]). This for instance allows the precise description of the domains of the powers of
which is used at various points in the argument. However, for the wave operators for the conic Laplacian the appropriate notion corresponds to the `edge' calculus of pseudodierential operators discussed originally by Mazzeo [9], arising from a ltration of the boundary (see also Schulze [15]). In this case, the manifold with boundary is X  R and the bers of the boundary are the surfaces t = const. Thus t is the base variable of the bration. To the edge calculus of pseudodierential operators, given by microlocalization from the dierential operators generated by xD , D (where the y's are tangential variables) and xD , we associate a notion of wavefront set. We can prove the propagation theorem analogous to that of Hormander for this `edge' wavefront set. However, in this new sense, D2 ,
is not globally of principal type but rather has two radial surfaces. These correspond to the end points of bicharacteristics arriving at, and leaving from, the boundary. At these surfaces there are restrictions on the propagation results, very closely related to those for scattering Laplacians in [10]. The construction of the test operator A; which away from the radial surfaces is essentially given by owout along the geodesic spray on @X; becomes more delicate at the radial surfaces. Positivity relies on the precise form of the singularity of the Hamilton vector eld there. These propagation estimates form the basis of both the diractive and geometric theorems. In the former we combine the estimates with a variant of the one-dimensional FBI transform, scaled with respect to the normal variable x. This reduces the diractive result to an iterative i

x

t

t

y

SINGULARITIES AND THE WAVE EQUATION ON CONIC SPACES

181

application of a uniqueness theorem for the Laplacian on the model, non-compact cone. To obtain the geometric theorem, showing that the outgoing singularities on non-geometrically related rays are weaker than the incoming ones, we use a division theorem. The additional hypothesis of microlocal tangential smoothing is shown to imply that the solution actually lies in a weighted Sobolev space with a higher x weight (hence more `divisible' by x) than is given, a priori, by energy conservation. This allows the microlocal propagation results indicated above to be pushed further at the outgoing radial surface and so yields the extra regularity. 8. Applications and extension The propagation of singularities results of the type discussed above should allow estimates of the spectral counting function as shown originally by Ivrii ([5], see also [12] and [3]). We expect these methods to extend to more complicated geometries, including manifolds with corners and iterated conic spaces.

References

[1] Je Cheeger and Michael Taylor, On the diraction of waves by conical singularities. I, Comm. Pure Appl. Math. 35 (1982), no. 3, 275{331, MR84h:35091a. [2] , On the diraction of waves by conical singularities. II, Comm. Pure Appl. Math. 35 (1982), no. 4, 487{529, MR84h:35091b. [3] L. Hormander, The analysis of linear partial dierential operators, vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. [4] Lars Hormander, On the existence and the regularity of solutions of linear pseudo-dierential equations, Enseignement Math. (2) 17 (1971), 99{163. MR 48 #9458 [5] V. Ivrii, On the second term in the spectral asymptotics for the LaplaceBeltrami operator on a manifold with boundary, Funct. Anal. Appl. 14 (1980), 98{106. [6] M. Kalka and A. Meniko, The wave equation on a cone, Comm. Partial Differential Equations 7 (1982), no. 3, 223{278, MR83j:58110. [7] G. Lebeau, Propagation des ondes dans les varietes a coins, Seminaire sur les E quations aux Derivees Partielles, 1995{1996, E cole Polytech., Palaiseau, 1996, MR98m:58137, pp. Exp. No. XVI, 20. [8] Gilles Lebeau, Propagation des ondes dans les varietes a coins, Ann. Sci. E cole Norm. Sup. (4) 30 (1997), no. 4, 429{497, MR98d:58183. [9] R. Mazzeo, Elliptic theory of dierential edge operators I, Comm. in P.D.E. 16 (1991), 1615{1664. [10] R.B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) (M. Ikawa, ed.), Marcel Dekker, 1994, pp. 85{130. [11] R.B. Melrose and J. Wunsch, Propagation of singularities for the wave equation on conic manifolds, In preparation.

182

RICHARD B. MELROSE AND JARED WUNSCH

[12] Richard Melrose, The trace of the wave group, Microlocal analysis (Boulder, Colo., 1983), Amer. Math. Soc., Providence, R.I., 1984, pp. 127{167. MR 86f:35144 [13] Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, A K Peters Ltd., Wellesley, MA, 1993. MR 96g:58180 [14] Michel Rouleux, Diraction analytique sur une variete a singularite conique, Comm. Partial Dierential Equations 11 (1986), no. 9, 947{988. MR 88g:58182 [15] B.-W. Schulze, Boundary value problems and edge pseudo-dierential operators, Microlocal analysis and spectral theory (Lucca, 1996), Kluwer Acad. Publ., Dordrecht, 1997, pp. 165{226. MR 98d:47112 Department of Mathematics, MIT, Cambridge MA 02139 Department of Mathematics, SUNY at Stony Brook, Stony Brook NY 11794

SOME REMARKS ON OSCILLATORY INTEGRALS GERD MOCKENHAUPT

1. Introduction The purpose of this note is to describe some results about oscillatory integral operators. Speci cally we are interested in bounds in Lebesgue spaces of operators given by T f (x) =

Z

Rk

ei'(x;) f ( ) d;

with '(x;  ) a real-valued smooth function on Rn  Rk ; k  n. Obviously T is bounded as maps from Lqcomp to Lploc. What is of interest here is the dependence of the norm for increasing . This will of course depend on the conditions we put on the phase function '. To guarantee that ' lives on an open subset of Rn  Rk it is natural to start with the condition (1) rank d dx' = k; x 2 Rn and  2 Rk : We will assume this condition throughout this note. For work related to weaker assumptions see, e.g. [21] and [18]. One of the questions we will ask is: What is the optimal (q; p)-range for which the operator T has norm of order ,n=p? In particular we would like to understand how this range will depend on k. To put things in perspective let us begin by describing what is known for the case k = n: A model phase function here is '(x;  ) = x
 , for x;  2 Rn. Then T is a localized version of the Fourier transform and the (Lqcomp; Lploc)-boundedness properties are covered by the HausdorYoung inequality. For general phase functions satisfying (1) the L theory of Fourier integral operators gives 2

C ,n=p; with p = q0  2 the dual exponent of q, i.e. 1=q0 + 1=q = 1. Next we consider the case k = n , 1: A basic result was obtained by E. M. Stein in the sixties. He discovered, for n  2, that the Fourier

kjTkjLqcomp!Lploc 

1991 Mathematics Subject Classi cation. 42B15. Key words and phrases. oscillatory integrals, restriction theorems. The support of the Australian Research Council is greatfully acknowledged. 183

184

GERD MOCKENHAUPT

transform has the following restriction property: For the unit sphere S n, in Rn and d a rotationally invariant measure 1

Z

(2)

jfb( )j

2

S ,

n 1

d ( )  C kf k2Lp0 (Rn ) ;

for some p0 > 1. By localizing to a ball of radius  in Rn, the dual of this inequality states that the operator T with phase function '(x;  ) = x
( ), where : U ! S n, parameterizes a coordinate neighborhood of the unit sphere in Rn, has norm of order ,n=p as a map from L (U ) to Lploc for some p < 1. Improvements on the range of exponents p were made by P. Tomas [26] and E.M. Stein. Moreover, it was shown by E.M. Stein (see [24]) that for nonlinear phase functions ' the norm of T has order ,n=p as an operator from Lcomp to Lploc for p  2(n + 1)=(n , 1), provided ' satis es the following curvature condition: for each x 2 Rn the hypersurface parameterized by (3)  7! rx '(x;  ) has nonvanishing Gaussian curvature. This (L ; Lp)-result is sharp in the sense that p = 2(n + 1)=(n , 1) is critical. Moreover, due to an example of J. Bourgain [2, 4] under the conditions (3) and (1) Stein's result can not be improved in case n is odd if we require q = 1 (see also [15]). However under some further p conditions, remarkable improvements have been made by J. Bourgain [1]. His method, which led to further improvements in [27, 5] n p = k and [25], showed in particular for the situation of the  QQ p = n,k k unit sphere described above, QQ QQ that for certain exponents Q p less then the critical L Q  q=2 q exponent 2(n +1)=(n , 1)n that Figure p; q k n k n, kTkL1 U !Lploc  C , p : It is expected that the (q; p)-range for which this inequality holds is determined by: p > 2n=k and p  (2n , k)=kq0 (see Figure). For n = 2 the norm of T is essentially well understood due to work of L. Carleson and P. Sjolin [6] provided the curvature condition (1) and (3) are satis ed. We note that for the expected bounds the crucial point is to understand for the operators T the (L1; Lp)-bounds. Our main concern here are the cases k < n , 1. There have been some results in the past addressing the problem of (Lqcomp; Lploc)-bounds 1

2

2

2

1

2

4

2

2

1

: (

)-range for

=

and

=

1.

(

)

SOME REMARKS ON OSCILLATORY INTEGRALS

185

for oscillatory integral operators in these cases. However, these results mainly discussed the cases k = 1; n , 2 or n=2 for n even (see e.g. [7], [8], [12], [11], [19], [20]). For dierent k some results are obtained in [10] and [17]. A natural question which we ask here is the following: Suppose (1) holds. Under which conditions on the phasenfunction ' does T map Lqcomp(Rk ) to Lploc (Rn) with norm of order , p in the full range 2n , k q0 and p > 2n p (4) for k < n? k

k

One of our results will be that we can expect these optimal bounds only when k  n=2. We will also see that in some situations where the phase function is linear in the x-variables an analogue of the SteinTomas result holds, i.e. optimal (L ; Lp)-bounds hold, but the (L1; Lp)bounds fail to hold in the range given in (4). This apparently appears only if k < n , 2. We should mention that one of the main diculties which distinguishes the case k < n , 1 from k = n , 1 lies in the fact that, although a stationary phase argument shows that for 2 C 1(Rk ) and most x 2 Rn the decay of T (x) is of order ,k= , in general isotropic bounds for Tf (x) decay slower (see e.g. [9]). 2

2

2. A necessary curvature condition Here we derive a necessary condition on the phase function ' such that T is bounded in the full range described in the above gure (for k  n). First we observe that if T has norm of order ,n=p, then for each x the operator with phase function 'R(x;  ) = R ('(x +x=R;  ), '(x ;  )) satis es the same bounds uniformly in R > 0. Hence, the operator with the linearized phase function (x;  ) 7! x
rx'(x ;  ) has the same bounds. By reparameterizing the k-dimensional submanifold  7! rx'(x ;  ) over the tangent plane at a given point using translation invariance we may assume that x
rx'(x ;  ) has the form (x
; x
( )), with (0) and d (0) both vanishing. A further scaling argument {replacing x ! Rx ; x ! R x and  ! =R and letting R ! 1{ shows that the phase function x
 + x
~( ), here ~ is the second order part of the Taylor expansion of , gives rise to an operator Z 2 T~f (x) = ei x1
 x2
 f ( ) e,jj = d; 0

0

0

0

0

0

1

2

1

1

2

2

2

1

R

(

k

+

~( ) )

2

2

which is bounded from Lq (Rk ) to Lp(Rn) for (q; p) on the line p = (2n , k)=kq0 provided that T has norm of order ,n=p on this line.

186

GERD MOCKENHAUPT

Write

~( ) = 1 
Q(x ); 2 P n,k with Q(x ) = j x ;j Bj and Bj 2 Sym(k), where Sym(k) denotes the space of symmetric matrices on Rk . To emphasize the dependence on Q the operator T~ will be denoted by TQ in the following and we refer to the submanifolds parameterized by H : Rk 3  7! (; 
B ; : : : ; 
Bn,k  ) 2 Rn as the associated quadratic submanifold MQ. If TQ maps L1(Rk ) to Lp(Rn), then in particular for the constant function 1 we have G = T~Q1 2 Lp. A computation gives x2


2

=1

2

2

1

kGk = C

(5)

p p

Z

R n,k

j det(E + iQ(x ))j,p= 2

2+1

dx2 ;

here E denotes the unit matrix in Sym(k). To ensure that the above integral is nite for some p < 1 we need that the symmetric matrices Bj ; 1  j  n , k, are linearly independent which requires that n  k(k + 3)=2. To nd a further restriction we show p n Proposition 2.1. If the function G above is in L (R ), for all p > 2n=k, then (6)

Z

S n,k ,1

j det Q(x)j, d(x) < 1 for all


0: Z Z1 rn,k,1 drd (y ); k (n,k )=k + S n,k,1 L(y) jr det Q(y )j which evaluates to (6) by integrating the inner integral. As a consequence we show: p n Corollary 2.2. Suppose the function G de ned above is in L (R ) for all p > 2n=k. Then the following hold:  If k is odd, then k  n2 .

SOME REMARKS ON OSCILLATORY INTEGRALS

187

 If k is even and the subspace fQ(x)jx 2 Rn,k g intersects the cone of positive de nite matrices in Sym(k), then k  n=2.

Proof. The idea here is to nd a hypersurface on the unit sphere in Rn,k where the function det Q vanishes at least of order 1. Assuming k < n= R 2, i.e. (n , k )=k > 1, Proposition (2.1) implies that the integral Sn,k,1 j det Qj,1 d is nite. Since the polynomial det Q(x) = det(x1 B1 +


+ xn,k Bn,k ), with Bi 2 Sym(k), is homogenous of degree k for some power  > 0 the function jxj j det Q(x)j,1 must be integrable over the unit ball in Rn,k . We can assume that det Q(x) does not vanish identically and that B1 is a diagonal matrix with entries 1. If k is odd we can write locally det Q(x) = (x1 , '(x2 ; : : : ; xn,k )) (x) where '; are real continuous functions and '(0) = 0. Hence , for all  > 0, j det Qj,1 is not locally integrable on the unit ball in Rn,k and therefore k  n=2. To show the second part we may assume that B1 = E . Then x1 ! Q(x1 ; x2 ; : : : ; xn,k ) is the characteristic polynomial of the symmetric matrix Q(0; x2; : : : ; xn,k ) and therefore has only real zeros. So again det Q(x) = (x1 , '(x2 ; : : : ; xn,k )) (x). As before we nd that k has to be  n=2. The condition in the proposition above may be phrased in an invariant way. Consider the submanifold M parameterized by  7! rx'(x0;  ) and x a point P = rx'(x0 ; 0). We assume that M carries the induced Euclidean metric. Let NP (M ) be the normal plane at P 2 M , TP (M ) be the tangent plane at P , v 2 NP (M ) and let GP (v ) be the Gaussian curvature at P of the orthogonal projection of M (along v ) into Rv  TP (M ). Then (6) states that

(7)

Z

S n,k,1 Np (M )

jGP (v)j, d(v) < 1 for all < n ,k k ;

where  denotes a nontrivial rotationally invariant measure on the unit sphere in NP (M ).

\

3. Restriction to quadratic submanifolds In the following we show some positive results for the operators TQ. We write TQf = f dQ, where dQ is the measure on Rn with support on MQ de ned by Q ( f ) =

Theorem 3.3. If

TQ is bounded

R

Z

j

f (; H ( )) e,jj =2 d: 2

Rk

j

, d (x) S n,k,1 det Q(x) 2 k p n from L ( ) to L ( ) for p

R

R

< 1 for = n,k k . Then

 2 nk,k . 2

188

GERD MOCKENHAUPT

It is enough to show (and in fact equivalent) that the compo0 c  f maps the dual space Lp (Rn ) into Lp (Rn ) for sition TQTQ f = d p  2 nk,k . Our strategy is now to de ne an analytic family Tz which evaluates at z = 0 to TQ and is bounded from L to L1 on the line 0 such that in the region s ,  < Re(s)  s the analytic singularities of Z (s) coincide with these of the zeta function 0

0

X X(,1) Z (s) = 1

0

m=0

0

mr T em(,sT + ) ;

where runs over the set of primitive periodic broken geodesics in , r = 0 if has an even number of re ection points and r = 1 otherwise, and  2 is determined by the spectrum of the linear Poincare map related to . The function Z (s) is rather similar to a dynamically de ned zeta function (cf. the survey of Baladi [Ba] for general information on this topic). One of the main tools to study this sort of zeta function is the so called Ruelle operator (well known e.g. from the study of Gibbs measures in statistical mechanichs and ergodic theory, cf. [R1] and [Si]). Ikawa [I4] succeeded to implement results of Pollicott (1986) and Haydn (1990) concerning the spectrum of the Ruelle operator and derived sucient conditions for Z (s) (and therefore Z (s)) to have a pole in a small neighbourhood of s in C. From this he derived the following. IR

0

0

0

Theorem 1. ([I4]) If K is a nite union of disjoint balls in

with the same radius  > 0 and  > 0 is suciently small, then LPC holds. 3 IR

The study of the scattering zeta function itself seems to be rather dif cult and very few results concerning it are known. Petkov [P] showed that Z (s) admits an analytic continuation in the domain Re(s)  s . Moreover, under some additional assumptions about the geometry of the obstacle K and assuming rational independence of the primitive periods of periodic orbits, Petkov proved that supt2 jZ (s + it)j = 1 (cf. [P] for some further results on the properties of Z (s)). Assuming that the broken geodesic ow has two periodic orbits and such that T 1 =T 2 is a diophantine number, Naud [N] proved that Z (s) has an analytic continuation to a domain of the form 0

0

IR

1

C jtj < Re(s)  s0 ; jtj = jIm(s)j  1 ; for some constants C > 0 and  > 0. s0 ,

2

204

LUCHEZAR STOYANOV

2. Spectrum of the Ruelle operator and dynamical zeta function

In the case when the obstacle K has the form (1), it is sometimes more convenient to work with the billiard ball map B (i.e. the shift along the broken geodesic ow from boundary to boundary) on the non ( n ) that wandering set M0 . The latter is the set of all points  2 S@K generate trapped broken geodesics in both directions. As a dynamical system B : M0 ,! M0 is naturally isomorphic to the Bernoulli shift on the symbolic space IR

1 Y  = f1 2 ,1

g

; ;::: ;s ;

the isomorphism being given by the natural coding of the geodesics by sequences fij g, where ij = 1; : : : ; s is the number of the connected component Ki containing the j th re ection point. Ikawa [I4] used the classical interpretation of the Ruelle operator as an operator acting on the space of Lipschitz functions on the symbol space . However, having in mind some signi cant recent developments, it seems more convenient to use a dierent model which is more closely related to the dynamics of the ow. In this section we describe some recent results of the author concerning the spectrum of the Ruelle operator and the dynamical zeta function related to the broken geodesic ow in the exterior of an obstacle of the form (1) in the 2. The results are similar to these of Dolgopyat [D] in the case of Anosov ows on compact manifolds. One would expect that similar results could be obtained (by using similar techniques) for obstacles in n, n  3. Such results would very likely lead to partial solutions of the LPC for cases much more general then the one considered in Theorem 1 above. From now on we assume that K is an obstacle of the form (1) in 2 . Let  be the non-wandering set of the broken geodesic ow in . Clearly  is the union of all orbits generated by elements of the set M0  ( ) and a suciently small  > 0 let de ned above. For x 2  n S@K s W (x) = fy 2 S  ( ) : (t (x); t (y ))   for all t  0 ; (t (x); t (y )) !t!1 0 g ; j

IR

IR

IR

( ) = fy 2 S ( ) : (t (x); t(y))   for all t  0 ; (t (x); t (y )) !t!,1 0 g be the (strong) stable and unstable sets of size . It is easy to show that  ( ) and every suciently small  > 0, W s(x) and for every x 2  n S@K  u

W x

SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 205 Wu (x) are 1-dimensional submanifolds of S  ( ). It is worth mentioning that both Ws(x)  and Wu(x)  are Cantor sets. Given  > 0, set

\

\

= fx = (q; v) 2 S ( ) : dist(q; @K ) > g ;  =  \ S( ) : In what follows in order to avoid ambiguity and unnecessary complications we will consider stable and unstable manifolds only for points x in S ( ) or ; this will be enough for our purposes. It follows from the genaral theory of horocycle foliations (cf. [S] or [KH]) that if  > 0 is suciently small, there exists  > 0 such that if x; y 2  and (x; y ) <  , then Ws (x) \  and  ,; (Wu (y ) \ ) intersect at exactly one point [x; y]. That is, there exists a unique t 2 [,; ] such that t ([x; y]) 2 Wu(x). Setting
(x; y) = t, de nes the so called temporary distance function
([Ch1]). If z 2  and U  Wu (z) and S  Ws(z) are closed (i.e. containing their end points) curves containing z and such that U \  and S \  have no isolated points, then  = [U \ ; S \ ] = f[x; y] : x 2 U \  ; y 2 S \  g is called a rectangle in . Notice that  is "foliated" by leaves [x; S \ ] of stable manifolds. Let R = fRigki be a family of rectangles such that each Ri is contained in a C cross-section Di  S( ) to the ow t . Thus, for each i, Ri = [Ui \ ; Si \ ], where Ui and Si are closed curves in Wu(zi) and Ws(zi), respectively, for some zi 2 . Set S ( )

[

1

]

=1

R

=

[ k

Ri :

i=1

The family R is called complete if there exists T > 0 such that for every x 2  there exist t 2 [,T; 0) and t 2 (0; T ] with t1 (x) 2 R and t2 (x) 2 R. Thus,  (x) = t (x) > 0 is the smallest positive time with P (x) =  x (x) 2 R, and P : R ,! R is the Poincare map related to the family R. Following [B] and [Ra] we will say that a complete family R = fRigki of rectangles in S( ) is a Markov family of size  2 (0; =2) for the billiard ow t if: (a) Ri \ Rj = ; for i 6= j and for each i the sets Ui \  and Si \  are contained in the interior of the curves Ui and Si, respectively; (b) diam(Ri )   ; (c) For any i 6= j and any x 2 Ri \ P , (Rj ) we have P ([x; Si \ ])  [P (x); Sj ] and P ([Ui \ ; x])  [Uj ; P (x)]; 1

2

2

( )

=1

1

206

LUCHEZAR STOYANOV

(d) For any i = 1; : : : ; k and x 2 Ri the function  is constant on the set [x; Si \ ]. The existence of a Markov family R of an arbitrarily small size  > 0 for t follows from the construction of Bowen [B] (cf. also Ratner [Ra]). It follows from a result of C. Robinson [Ro] that there exists an open neighbourhood V of  in S ( ) and C transverse foliations F u and F s on V such that Wu(x) \ V = F u (x) and Ws(x) \ V = F s(x) for any x 2  \ V . Fix a neighbourhood V and C foliations F u and F s with these properties. Choosing  > 0 suciently small, we may assume that our Markov family R satis es the following additional condition: 1

1

(e) For each i the cross-section Di (and therefore the rectangle Ri) is contained in V . In what follows we assume that R = fRigki is a xed Markov family for t satisfying the extra conditions (e) and (f). Then U

[ = k

=1

Ui ;

i=1

is a nite disjoint union of compact curves in V . Using the foliations F u and F s in V , assuming again that V is suciently small, we can extend the product [x; y] over the whole Ui  Si for any i as follows. Given x 2 Ui  Wu(zi) = F u(zi) \ V and y 2 Si  Ws(zi ) = F s(xi ) \ V , the sets F s (x) and  ,; (F u(y )) intersect at exactly one point [x; y]. The image of the C map Ui  Si 3 (x; y) 7! [x; y] is a 2-dimensional sumanifold of S ( ) which will be denoted by R~i = [Ui; Si]. The pro~ = [ki R~i ,! U along the leaves of F s is C . Then the jection p : R Poincare map P : R~ ,! [ki Di and the corresponding root function  : R~ ,! [0; 1) are well-de ned and C . Thus,  can be extended to a C map  : U ,! U by the same formula:  = p  P . In the same way one observes that
(x; y) is well-de ned and C for (x; y) 2 [ki R~i R~i . Let C (U ) be the space of bounded continuous functions on U . For g 2 C (U ) denote kg k = kg k = supx2U jg (x)j. Given g , the Ruelle operator Lg : C (U ) ,! C (U ) is de ned in the usual way: [

]

1

1

=1

=1

1

1

1

0

(Lg h)(u) =

X

 (v )=u

eg(v) h(v ):

=1

SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 207

If g 2 C 1 (U ), then clearly Lg preserves the space C 1(U ). Here C 1 (U ) denotes the space of dierentiable function with bounded derivatives on U . A conjugacy between  \ R and the Bernoulli shift on a symbolic space A is now de ned in the usual way. Let A = (Aij )ki;j=1 be the matrix given by Aij = 1 if P (Ri) \ Rj 6= ; and Aij = 0 otherwise. Then A = f(ij )1 j =,1 : 1  ij  k; Aij ij+1 = 1 for all j g; and the Bernoulli shift ~ : A ,! A is given by ~ ((ij )) = ((i0j )), where i0j = ij+1 for all j . De ne  : R ,! A by (x) = (ij )1 j =,1, where P j (x) 2 Rij for all j 2 Z. Notice that P : R ,! R is invertible, so P j is well-de ned for all integers j . The map  is a homeomorphism when A is considered with the product topology, and ~   =   P . The projection r of  on A is given by r   =  . The subset  \ U of R can be naturally identi ed with +A = f (ij )1 j =0 : 1  ij  k ; Aij ij+1 = 1 for all j  0 g: Namely, if  : A ,! +A is the natural projection, then + =   j\U :  \ U ,! +A is a bijection with ~  + = +  , where ~ denotes the corresponding Bernoulli shift on +A . The dynamical zeta function of the broken geodesic ow t is de ned by Y  (s) = (1 , e,s`( )),1 ;

where runs over the set of closed orbits of t and `( ) is the least period of . One can easily see that

0 1 1 X X  (s) = exp B @ n1 n e,sn(x)CA ; n=1

where

 (x)=x x2U \

n(x) =  (x) +  ((x)) + : : : +  (n (x))

is the period (length) of the closed orbit generated by x. That is, we have ! 1 X 1  (s) = exp Zn(,s ) ; where

n=1

Zn(,s ) =

n

X

n (x)=x x2U \

e,sn (x) :

208

LUCHEZAR STOYANOV

The following lemma of Ruelle ([R2], cf. also [PoS]) partially explains the relationship between the Ruelle operator and the behaviour of the dynamical zeta function  (s). 1

j

Lemma 1. (Ruelle) Let xj be an arbitrary point in Uj \  for every = 1; : : : ; k. There exist constants C > 0,  > 0,  2 (0; 1) such that Zn ( s )

,

,

k , X j =1

n

L,s Uj




(xj )

 C jIm(s)jnn

for all n  1 and all s 2 C with Re(s)  hT , , hT being the topological entropy of the ow t , and U is the characteristic function of Uj in U . j

The suspended ow ~ r over rA = f(; t) :  2 A; 0  t  r( ) g is de ned by ~sr (; t) = (; t + s), where we use the identi cation (; r( )) = (( ); 0) in A  . Then r;A = f(; t) 2 rA :  2 A g is a closed subset of rA which is invariant under the semi ow ~tr , t  0 (cf. [PP2] for details). A conjugacy between ~tr and t on  is de ned by r (s(x)) = ((x); s) for x 2 R and 0  s   (x). Given  2 (0; 1), consider the metric d on A given by d (; ) = 0 if  =  and d (;  ) =  n if i = i for jij  n and n is maximal with this property. In a similar way one de nes a metric d on A . Using it, we get a metric on r;A by setting dr ((; t); (; s)) = d (; )+ jt , sj. The spaces of Lipschitz functions on A and r;A with respect to the metrics just de ned will be denoted by F (A ) and F (r;A ), respectively. In the present setting the well-known Perron-Ruelle-Frobenius theorem reads as follows. +

IR

+

+

+

+

+

+

+

Perron-Ruelle-Frobenius Theorem. Let f 2 F (A ) be a realvalued function. (a) The Ruelle operator Lf : F (A ) ,! F (A ) has a simple eigen+

+

+

value  = ePr(f ) , where Pr(f ) is the topological pressure of f , and a strictly positive eigenfunction h 2 F (+A ). (b) spec(Lf ) n fg is contained in a disk of radius strictly less than . (c) There exists a unique probability measure  = f on +A such that Z

()

Lf g d

=

Z

g d

for all g 2 F (+A ). 1

Its statement is slightly modi ed to suit the present context.

SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 209

The measure f is the so called Gibbs measure related to the potential ([R1], [Si]). The complex case is more complicated. It was estabslished by Pollicott [Po] that for any f = u + iv 2 F (A ) the spectral radius of Lf as an operator on F (A ) is not greater than ePr u . Moreover, if Lf has an eigenvalue  with jj = ePr u , then  is simple and unique and the rest of the spectrum is contained in a disk of strictly smaller radius. If Lf has no eigenvalues  with jj = ePr u , then the whole spectrum of Lf is contained in a disk of radius less than ePr u . The following result was obtained by using a modi cation of the technique developed by Dolgopyat [D] in order to prove a similar result in the case of Anosov ows on compact manifolds , in particluar for geodesic ows on surfaces of negative curvature. f

+

+

( )

( )

( )

( )

2

Theorem 2. ([St2]) There exist constant c0 < hT and  2 (0; 1) such that for Re(s)  c0 and jIm(s)j >> 0 the spectral radius of the Ruelle operator L,s does not exceed . Using the above theorem and applying the argument of Pollicott and Sharp [PoS] in the case of geodesic ows on compact surfaces of negative curvature, one derives that the zeta function Y  (s) = (1 , e,s` ), ( )

1

of the billiard ow t has an analytic continuation in a half-plane Re(s) > c for some c < hT except for a simple pole at s = hT . Moreover, folowing [PoS] again, one derives that there exists c 2 (0; hT ) such that  () = #f : `( )  g = li(ehT  ) + O(ec) R as  ! 1, where li(x) = x du= log u. The latter is a much stronger result than the standard Prime Orbit Theorem for open planar billiards in [Mor] (cf. [St1] for the higher dimensional case) derived by means of a result of Parry and Pollicott [PP1]. 0

0

2

3. Exponential decay of correlations In this section we continue to consider the case when K is an obstacle of the form (1) in , and we also use most of the notation from Sect.2. 2 IR

In fact the primary aim of Dolgopyat was to establish exponential decay of correlations for such ows. See Sect.3 for more information. 2

210

LUCHEZAR STOYANOV

Let F 2 F () for some  > 0. Denote by F~ the function on rA such that F~  r = r  F . There exists  = () 2 (0; 1) such that F~ 2 F (rA). Let ~ be the Gibbs measure related to F~ with respect to the suspended ow ~r , and let P = Pr~r (F~ ) be the topological pressure of F~ with respect to ~ r . A function f~ 2 F (+A ) related to F~ is de ned by Z r() ~ f ( ) = F~ (; s) ds : 0

Let ~ be the Gibbs measure on A determined by the function f~ , Pr. Then (cf. [PP2]) Z 1 d~(; s) =  (r) d~( )ds ; where  (~g) = g~( ) d~( ) : A Moreover, we have Pr~ (f~ , Pr) = 0: The Gibbs measures ~ and ~ give rise to measures  and  on R and  \ U , respectively, via the conjugacies r and . IfZf is the function on  \ U such that f~   (x)  =   f , then f (x) = F (s(x)) ds, so f 2 F ( \ U ). The 0 measures  and  are called the Gibbs measures related to F and f ,P , respectively. It follows from above that Prt (F ) = P and Pr (f ,P ) = 0. Given a a Holder continuous potential F on  and arbitrary A; B 2 F(), the correlation function of A and B is de ned by

Z

Z

A;B (t) = A(x)B (t(x)) dF (x), 

Z



A(x) dF (x)





B (x) dF (x) :

It is an important problem in smooth ergodic theory (and also in various areas in physics) to know whether such a function decyas exponentially fast as t ! 1. Using Theorem 2 above and the technique developed by Dolgopyat [D], one immediately gets the following.

Theorem 3. Let F be a Holder continuous function on  in S ( )

and let F be the Gibbs measure determined by F on . For every  > 0 there exist constants C = C () > 0 and c = c() > 0 such that jA;B (t)j  Ce,ctkAk kB k for any two functions A; B 2 F ().

Here khk denotes the Holder constant of h 2 F ().

SPECTRUM OF THE RUELLE OPERATOR AND ZETA FUNCTIONS... 211

We refer the reader to the recent survey article of Baladi [Ba] for general information and historical remarks on decay of correlations. Amongst the most recent achievements one should mention the important articles of Liverani [L], Young [Y], Chernov [Ch1] and Dolgopyat [D] answering long standing questions. It apppears that for billiards the only results of this type that have been known so far concern the corresponding discrete dynamical system (generated by the billiard ball map from boundary to boundary). To my knowledge, these results are: the subexponential decay of correlations established for a very large class of dispersing billiards by Bunimovich, Sinai and Chernov [BSC] and the exponential decay of correlations for some classes of dispersing billiards in the plane and on the two-dimensional torus established by Young [Y] and Chernov [Ch2] as consequences of their more general arguments. Theorem 3 above describes a non-trivial class of billiard (broken geodesic) ows with exponential decay of correlations for any Holder continuous potential.

References

[Ba] V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys. 18 (1998), 255-292. [B] R. Bowen, Symbolic dynamics for hyperbolic ows, Amer. J. Math. 95 (1973), 429-460. [BSC] L. Bunimovich, Ya. Sinai,and N. Chernov, Statistical properties of twodimensional hyperbolic billiards, Russ. Math. Surveys 46 (1991), 47-106. [Ch1] N. Chernov, Markov approximations and decay of correlations for Anosov

ows, Ann. of Math. 147 (1998), 269-324. [Ch2] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94 (1999), [D] D. Dolgopyat, On decay of correlations in Anosov ows, Ann. of Math. 147 (1998), 357-390. [G] Gerard C.: Asymptotique des p^oles de la matrice de scattering pour deux obstacles strictement convexes. Bull. de S.M.F., Memoire n 31, 116 (1988). [GM] V.Guillemin and R.Melrose, The Poisson sumation formula for manifolds with boundary, Adv. in Math. 32 (1979), 204-232. [I1] M.Ikawa, On the poles of the scattering matrix for two strictly convex obstacles, J. math. Koyto Univ. 23 (1983), 127-194. [I2] M. Ikawa, Decay of solutions of the wave equation in the exterior of several strictly convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146. [I3] M. Ikawa, On the distribution of poles of the scattering matrix for several convex bodies, pp. 210-225 in Lecture Notes in Math., vol. 1450, Springer, Berlin, 1990. [I4] M. Ikawa, Singular perturbations of symbolic ows and poles of the zeta functions, Osaka J. Math. 27 (1990), 281-300; Addendum: Osaka J. Math. 29 (1992), 161-174. [KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge 1995.

212

LUCHEZAR STOYANOV

[LP] P. Lax, R. Phillips, Scattering Theory, Revised Ed., Academic Press, London, 1989. [L] C. Liverani, Decay of correlations, Ann. of Math. 142 (1995), 239-301. [M] R. Melrose: Geometric Scattering Theory, Cambridge Univ. Press, 1994 [MS] R. Melrose, J. Sjostrand: Singularities in boundary value problems. Comm. Pure Appl. Math. 31 (1978), 593-617; 35 (1982),129-168. [Mor] T. Morita, The symbolic representation of billiards without boundary condition, Trans. Amer. Math. Soc. 325 (1991), 819-828. [N] F. Naud, Analytic continuation of the dynamical zeta function under a diophantine condition, Preprint 2000. [PP1] W. Parry and M. Pollicott, An analogue of the prime number theorem and closed orbits of Axiom A ows, Ann. Math. 118 (1983), 573-591. [PP2] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque 187-188, 1990. [P] V. Petkov, Analytic singularities of the dynamical zeta function, Nonlinearlity 12 (1999), 1663-1681. [Po] M. Pollicott, On the rate of mixing of Axiom A ows, Invent. Math. 81 (1985), 413-426. [PoS] M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042. [Ra] M. Ratner, Markov partitions for Anosov ows on n-dimensional manifolds, Israel J. Math. 15 (1973), 92-114. [Ro] C. Robinson, Structural stability of C 1 ows, Lect. Notes in Math. 468 (1975), 262-277 [R1] D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading, Mass., 1978. [R2] D. Ruelle, Resonances for Axiom A ows, Commun. Math. Phys. 125 (1989), 239-262. [Si] Ya. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 27 (1972), 21-69. [S] S. Smale, Dierentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. [St1] L. Stoyanov, Exponential instability and entropy for a class of dispersing billiards, Ergod. Th. & Dynam. Sys. 19 (1999), 201-226. [St2] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard ows, Amer. J. Math., to appear. [Y] L.-S. Yang, Statistical properties of systems with some hyperbolicity including certain billiards, Ann. Math. 147 (1998), 585-650. [Z] M. Zworski, Counting scattering poles, In: Spectral and Scattering Theory, M. Ikawa Ed., Marcel Dekker, New York 199 ; pp. 301-331. Luchezar Stoyanov, Department of Mathematics and Statistics, University of Western Australia, Perth WA 6709, Australia

E-mail address :

[email protected]

ON THE BANACH-ISOMORPHIC CLASSIFICATION OF p SPACES OF HYPERFINITE SEMIFINITE VON NEUMANN ALGEBRAS L

F. A. SUKOCHEV

Abstract. We present a survey of recent results in the Banach space classi cation of non-commutative Lp -spaces.

An important role in Banach space theory has always been played by the problem of classifying Banach spaces. This problem has many facets. In our present setting we address this problem by looking at a Banach space as a linear topological space. The natural maps then are continuous linear operators and we look for invariants under isomorphism (=bicontinuous one-to-one linear operator). In general, the development of Banach space theory has clearly shown that there is no hope left for a complete structural theory of Banach spaces, although one can still hope to have such a theory in some special cases. Our objective in the present talk is to describe recent results in this direction in the special case of non-commutative pspaces associated with semi nite von Neumann algebras. To place this work in its proper context we brie y review its origins, beginning with the work of both the mathematicians mentioned so far: Banach and von Neumann. Many fruitful directions in Banach space theory emerged from the famous book [B] by Banach, and the date of appearance of the French edition of this book (1932) is usually regarded as the date of birth of the theory itself. The nal chapter (XII) of this book discusses in depth the problems of comparison between the elements of the two families of Banach spaces (perhaps, the most important families of classical Banach spaces): the spaces p and p = p(0 1), 1  1. Recall that p is the space of all in nite complex-valued sequences = ( n)1 n=1 , such that 1 k klp := ( j njp)1=p 1 L

l

l

a

L

X

L

;

p
Cn 3 j p 2 : 1 1

,1

The following consequence is straightforward.

Corollary 3 [M], (1967) There exists an isomorphic embedding of

the space Cp into Lp if and only if p = 2.

The converse to the result of Corollary 3 was obtained by Arazy and Lindenstrauss [AL].

Theorem 4 [AL], (1975) There exists an isomorphic embedding

of the space Lp into Cp if and only if p = 2.

216

F. A. SUKOCHEV

One by-product of the Arazy-Lindenstrauss arguments was an identi cation of yet another member of the p -family, the spaces p 1  1. The de nition is very simple 1 n p := (n=1 p )p i.e. each element 2 p is represented by an in nite sequence ( n )1 n=1 with n 2 pn and 1 k kSp := ( k nkCpn )1=p L

S ;

p