Integrable Hamiltonian systems and spectral theory 5939722741, 9785939722742

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C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

J¨urgen Moser

Integrable Hamiltonian Systems and Spectral Theory

Moscow Izhevsk 2003

C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

Published by Regular and Chaotic Dynamics, Moscow-Izhevsk Universitetskaya, 1, Izhevsk, Russia, 426034 Phone: (7–3412) 50–02–95 Fax: (7–3412) 50–02–95 E-mail: [email protected]

Acknowledgement. The publisher is grateful to Springer-Verlag for the permission to reprint the papers included in this volume.

Moser, J¨urgen INTEGRABLE HAMILTONIAN SYSTEMS AND SPECTRAL THEORY c 2003 by Regular and Chaotic Dynamics, Moscow-Izhevsk  c 2003 by Institute of Computer Science, Moscow-Izhevsk 

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ISBN 5-93972-274-1

Printed on acid-free paper Printed in the Russian Federation

C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

Contents

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Editorial Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Finitely Many Mass Points on the Line under the Influence of an Exponential Potential — an Integrable System . . . . . . . . . . . . . . § 1. Analogue of the Toda Lattice for Finitely Many Mass Points . . . § 2. Flaschka’s Form of the Differential Equation and Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Partial Fractions and Continued Fractions . . . . . . . . . . . . . § 4. Solution of the Scattering Problem . . . . . . . . . . . . . . . . . § 5. Associated Differential Equations . . . . . . . . . . . . . . . . . Three Integrable Hamiltonian Systems Connected with Isospectral Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Isospectral Deformations . . . . . . . . . . . . . . . . . . . . . . § 3. The n-Particle System on the Line with the Inverse Square Potential § 4. Asymptotic Behavior, Marchioro’s Conjecture . . . . . . . . . . § 5. The Periodic Case — Sutherland’s Equation . . . . . . . . . . . . § 6. Rational Character of the Solution of (2.4) . . . . . . . . . . . . § 7. The Scattering Problem Associated with the Equation of Kac and Van Moerbeke . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Aspects of Integrable Hamiltonian Systems . . . . . . . . . . § 1. Integrable Hamiltonian Systems . . . . . . . . . . . . . . . . . . § 2. Examples of Integrable Systems, Isospectral Deformations . . . . § 3. Reduction of a Hamiltonian System with Symmetries . . . . . . . § 4. The Inverse Square Potential . . . . . . . . . . . . . . . . . . . . § 5. Extension of the Geodesic Flow . . . . . . . . . . . . . . . . . . § 6. Geodesics on an Ellipsoid . . . . . . . . . . . . . . . . . . . . . § 7. An Integrable System on the Sphere . . . . . . . . . . . . . . . . § 8. Hill’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 21 27 34 41 41 45 47 50 53 56 60 65 65 68 71 80 89 96 102 110

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Geometry of Quadrics and Spectral Theory . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a. Background . . . . . . . . . . . . . . . . . . . . . . . . . b. Geodesics on an Ellipsoid . . . . . . . . . . . . . . . . . c. Perturbations of Rank 2 . . . . . . . . . . . . . . . . . . . d. Hyperelliptic Curve . . . . . . . . . . . . . . . . . . . . . e. Applications . . . . . . . . . . . . . . . . . . . . . . . . . f. Connection with M. Reid’s Result [15] . . . . . . . . . . . g. Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . § 2. Perturbation of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . a. Isospectral Manifolds . . . . . . . . . . . . . . . . . . . . b. Isospectral Deformations . . . . . . . . . . . . . . . . . . c. The Action of Gl(2, R) . . . . . . . . . . . . . . . . . . d. Trace Formulae . . . . . . . . . . . . . . . . . . . . . . . § 3. Connection with Confocal Quadrics . . . . . . . . . . . . . . . . a. Integrals for the Geodesic Flow on the Ellipsoid . . . . . b. Isospectral Deformation . . . . . . . . . . . . . . . . . . c. Interpretation of the Eigenvalues and the Frame of L . . . d. Joachimsthal’s Integral . . . . . . . . . . . . . . . . . . . § 4. The Hyperelliptic Curve . . . . . . . . . . . . . . . . . . . . . . a. The Isospectral Manifold M(λ) . . . . . . . . . . . . . . b. An Inverse Spectral Problem . . . . . . . . . . . . . . . . c. The Symplectic Structure . . . . . . . . . . . . . . . . . . d. Degenerate Case . . . . . . . . . . . . . . . . . . . . . . e. Limit Cases . . . . . . . . . . . . . . . . . . . . . . . . . § 5. Examples of Integrable Flows . . . . . . . . . . . . . . . . . . . a. Constrained Systems . . . . . . . . . . . . . . . . . . . . b. A Mass Point on the Sphere S n−1 : |x| = 1 under the Influence of the Force −Ax (C. Neumann [14]) . . . . . . c. A Mass Point on the Ellipsoid Q0 (x) + 1 = 0 under the Influence of the Force −ax (Jacobi [6]) . . . . . . . . . . d. Geodesic Flow on the Orthogonal Group (Manakov [8], Mischenko [11]) . . . . . . . . . . . . . . . . . . . . . . e. Hill’s Equation (McKean and Trubowitz [9, 10]) . . . . . § 6. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 123 124 127 129 129 130 130 131 131 133 137 138 140 140 143 145 147 148 148 152 156 159 160 162 162 164 165 167 168 171

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Contents

Integrable Hamiltonian Systems and Spectral Theory . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Classical Integrable Hamiltonian Systems and Isospectral Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Hamiltonian systems . . . . . . . . . . . . . . . . . . . . 2. Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Perturbation of integrable systems . . . . . . . . . . . . . 4. The inverse square potential . . . . . . . . . . . . . . . . 5. Constrained Hamiltonian systems . . . . . . . . . . . . . § 3. Geodesics on an Ellipsoid and the Mechanical System of C. Neumann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Geodesic flow on the ellipsoid . . . . . . . . . . . . . . . 2. Confocal quadrics, construction of integrals . . . . . . . . 3. Isospectral deformations . . . . . . . . . . . . . . . . . . 4. The mechanical problem of C. Neumann . . . . . . . . . . 5. The connection between the two systems via the Gauss mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Riemann surface . . . . . . . . . . . . . . . . . . . . § 4. The Schr¨odinger Equation for Almost Periodic Potentials . . . . 1. The spectral problem . . . . . . . . . . . . . . . . . . . . 2. The periodic case . . . . . . . . . . . . . . . . . . . . . . 3. Almost periodic potential . . . . . . . . . . . . . . . . . . 4. The rotation number . . . . . . . . . . . . . . . . . . . . 5. The Green’s function and a trace formula . . . . . . . . . 6. Connection with the KdV equation . . . . . . . . . . . . . § 5. Finite Band Potentials . . . . . . . . . . . . . . . . . . . . . . . 1. Formulation of the problem . . . . . . . . . . . . . . . . 2. Representation of G(x, x; λ) in terms of partial fractions 3. Connection with the mechanical problem . . . . . . . . . 4. Solution of the inverse problem . . . . . . . . . . . . . . 5. Finite gap potentials as almost periodic functions . . . . . 6. The elliptic coordinates on the sphere . . . . . . . . . . . 7. Alternative choice of the branch points . . . . . . . . . . § 6. Limit Cases, Bargmann Potentials . . . . . . . . . . . . . . . . . 1. Schwarz – Christoffel mapping . . . . . . . . . . . . . . . 2. Basis for the frequency module . . . . . . . . . . . . . . 3. Stationary solutions and their stability behavior . . . . . . 4. The flow on the unstable manifold W+ (en ) . . . . . . . .

5 175 175 179 179 180 183 184 186 189 189 191 193 194 195 199 202 202 203 206 207 208 211 214 214 215 217 219 221 223 224 225 225 226 228 229

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5. 6. 7. 8.

The Bargmann potentials . . A focussing property on S 2 N -solitons . . . . . . . . . . Concluding remarks . . . . .

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231 234 236 237

Discrete Versions of Some Classical Integrable Systems and Factorization of Matrix Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 241 § 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 § 1. The Discrete Version of the Dynamics of a Rigid Body . . . . . . 246 1.1. The Equations of «Motion» . . . . . . . . . . . . . . . . 246 1.2. The Solution of the Matrix Eq. (6): ω T J − Jω = M . . 249 1.3. Isospectral Deformations . . . . . . . . . . . . . . . . . . 252 1.4. The Symplectic Geometry of Eq. (6) . . . . . . . . . . . 254 1.5. The Integration of the Discrete Euler Equation . . . . . . 258 1.6. Explicit Formulas for the Discrete Dynamics of the 3-Dimensional Rigid Body . . . . . . . . . . . . . . . . . . . 261 § 2. The Discrete Dynamics on Stiefel Manifolds and the Heisenberg Chain with Classical Spins . . . . . . . . . . . . . . . . . . . . . 264 2.1. The Equation of the Dynamics and Isospectral Deformations265 2.2. Discrete Version of the Neumann System and the Heisenherg Chain With Classical Spins . . . . . . . . . . . . . . 266 § 3. The Billiard Inside an Ellipsoid . . . . . . . . . . . . . . . . . . 269 3.1. The Splittings and Isospectral Deformations . . . . . . . . 270 3.2. Connection Between the Ellipsoidal Billiard and the Discrete Neumann System . . . . . . . . . . . . . . . . . . . 272

C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

Curriculum Vitae Personal Data Born: July 4, 1928, K¨onigsberg, Germany; U S Citizen Education 1947 – 1952 1952

Student at the University of G¨ottingen, Germany Dr. rer. nat. (Ph. D.), University of G¨ottingen Employment and Professional History

1953 – 1954 1954 – 1955 1955 – 1956 1956 – 1957 1957 – 1960

Fulbright Fellowship, to visit New York University Assistant (with C. L. Siegel) in G¨ottingen Research Associate, New York University Assistant Professor, New York University Associate Professor, M. I. T. (Massachusetts Institute of Technology), Cambridge

C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

8 1960 – 1980 1960 – 1967 1961 – 1963 1967 – 1970 1980 – 1995 1983 – 1986 1984 – 1995 1991 – 1997

Curriculum Vitae

Professor, Courant Institute of Mathematical Sciences, New York Consultant to IBM, Yorktown Heights Sloan Fellowship, to visit USSR Director of the Courant Institute, New York Professor Eidgen¨ossische Technische Hochschule, ETH Z¨urich, Switzerland President of the International Mathematical Union (IMU) Director of the Mathematics Research Institute, ETH Z¨urich (Forschungsinstitut f¨ur Mathematik – FIM) Obmann der Sektion (reine) Mathematik der Deutschen Akademie der Naturforscher Leopoldina Memberships

American Mathematical Society Society for Industrial and Applied Mathematics (SIAM) International Astronomical Union (IAU) – Consultant American Academy of Arts and Sciences, Cambridge/MA (1964) National Academy of Sciences of the USA (1971) Akademie der Wissenschaften und der Literatur, Mainz, Germany (1981) The Royal Swedish Academy of Sciences (1981) Deutsche Akademie der Naturforscher Leopoldina, Halle, Germany (1982) International Mathematical Union, President (1983 – 1986) The Finnish Academy of Science and Letters (1987) Российская академия наук (1994) Acad´emie des Sciences de la France (1995) Московское Математическое Общество (1995) The London Mathematical Society (1996) Schweizerische Mathematische Gesellschaft (1997) Awards George D. Birkhoff Prize in Applied Mathematics (1968) Craig Watson Medal, National Academy of Sciences, USA (1969) J. von Neumann Lecture, SIAM, Seattle (1984) L. E. J. Brouwer Medal, Groningen (1984) Professor Hon´orario, IMPA, Rio de Janeiro (1989) Dr. rer. nat. h. c., Ruhr-Universit¨at Bochum (1990) Doctor h. c., Universit´e Pierre et Marie Curie de Paris (1990)

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Georg-Cantor-Medaille, DMV-Tagung Berlin (1990) Wolf Prize (Wolf Foundation Israel) (1994/95)

Publications 1) St¨orungstheorie des kontinuierlichen Spektrums f¨ur gew¨ohnliche Differentialgleichungen zweiter Ordnung. Math. Ann. 125, 1953, 366–393. ¨ 2) Uber periodische L¨osungen des restringierten Dreik¨orperproblems, die sich erst nach vielen Uml¨aufen schliessen. Math. Ann 126, 1953, 325–335. ¨ 3) Uber periodische L¨osungen kanonischer Differentialgleichungssysteme. Nachrichten der Akademie der Wissenschaften, G¨ottingen, Math. Phys. Kl. IIa, 1953, 23–48. 4) Singular perturbation of eigenvalue problems for linear differential equations of even order. Comm. Pure Appl. Math. 8, 1955, 251–278. 5) Nonexistence of integrals for canonical systems of differential equations. Comm. Pure Appl. Math. 8, 1955, 409–436. 6) Stabilit¨atsverhalten kanonischer Differentialgleichungssysteme. Wiss. G¨ottingen. Math. Phys. Kl. IIa, 1955, 87–120.

Nachr.

Akad.

7) The resonance lines for the synchroton. Proc. of the CERN Symposium, I, 1956, 290–292. 8) Analytic invariants on an area-preserving mapping near an unstable fixed point. Comm. Pure Appl. Math. 9, 1956, 673–692. 9) On the generalizations of a theorem of A. Liapounoff. Comm. Pure Appl. Math., 11, 1958, 257–271. 10) New aspects in the theory of stability of Hamiltonian systems. Comm. Pure Appl. Math., 11, 1958, 81–114. 11) Stability of the Asteroids. The Astronomical Journal, 63, 1958, 439–443. 12) On the elimination of the irrationality condition and Birkhoff’s concept of complete stability. Boletin de la Soc. Mat. Mexicana, 1960, 167–175. 13) On the integrability of an area-preserving Cremona mappings near an elliptic fixed point, Boletin de la Soc. Mat. Mexicana, 1960, 176–180. 14) Remarks on the preceding paper of Louis Howard. Journal of Math. Phys., 37, 1959, 299–304. 15) A new proof of di Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Appl. Math., Vol. 13, 1960, 457–468.

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16) Bistable system of differential equations. Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations. Proc. of the Rome Symposium (Sept. 1960), organized by the Prov. Internat. Computation Centre, Birkh¨auser Verlag, Basel, 1960, 320–329. 17) The order of a singularity in Fuchs’ theory. Math. Zeitschrift, 72, 1960, 379–398. 18) Bistable systems of differential equations with applications to tunnel diode circuits. IBM Journal of Research and Development, Vol. 5, No.3, 1961, 226–240. 19) A new technique for the construction of solutions for nonlinear differential equations. Proc. Nat. Acad. of Sci., USA, Vol. 47, No. 11, 1961, 1824–1831. 20) On the regularity problem for elliptic and parabolic differential equations. Proceedings of an International Conference on Partial Differential Equations and Continuum Mechanics. The Univ. of Wisconsin Press, 1961, 159–169. 21) On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math., 14, 1961, 577–591. 22) Stability and nonlinear character of ordinary differential equations. Nonlinear problems. Proc. Symposium, Madison, Wisconsin, April 30 – May 1, 1962, Ed. Langer. The University of Wisconsin Press 1963, 139–150. 23) On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. G¨ottingen, Math. Phys. Kl. IIa, 1962, 1–20. 24) New results on the stability of periodic motions. Proc. Internat. Congress Math., Stockholm 1962, 584–586. 25) Perturbation theory for almost periodic solutions for undamped nonlinear differential equations. Internat. Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Acad. Press, 1963, 71–79. 26) On the differential equations of electrical circuits and the global nature of the solutions. Internat. Symposium on Nonlinear Diff. Equations and Nonlinear Mechanics. Acad. Press, 1963, 147–154. 27) Some problems and results in the theory of nonlinear differential equations. Proc. IBM Scientific Computing Symposium. Dec. 9–11, 1963; Data Proc. Division, White Plains, N.Y., 1965, 5–16. 28) On invariant manifolds of vector fields and symmetric partial differential equations. Differential Analysis, Bombay Coll., 1964, 227–236. 29) On the volume elements on a manifold. Transactions of the AMS, 120, No. 2, 1965, 286–294. 30) Some results in the stability of nonlinear networks containing negative resistances. IEEE Trans. on Circuit Theory. Corresp., Vol. CT-11, No. 1, 1964, 165–167.

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31) A theory of nonlinear networks (with R. K. Brayton). I: Quarterly of Appl. Math. 22, No. 1, 1964, 1–33 II: Quarterly of Appl. Math. 22, No. 2, 1964, 81–104. 32) I: Reprinted in Nonlinear Networks: Theory and Analysis, ed. Alan N. Wilson, Jr., JEEE Press, New York 1975, 132–164. 33) A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math., 17, 1964, 101–134. 34) Combination tones for Duffing’s equation. Comm. Pure Appl. Math. 18, 1965, 167–181. 35) A rapidly convergent iteration method and nonlinear differential equations. I: Scuola Normale Sup. Pisa Ser. III, Vol. 20, 1966, 265–315 II: Scuola Normale Sup. Pisa, Ser. III, Vol. 20, 1966, 499–535. 36) A rapidly convergent iteration method and nonlinear differential equations I, II. Russian Translation of [34]. Uspekhi Mat. Nauk 23, 1968, 179–328. 37) Quasi-periodic solutions for the three-body problem (with W. H. Jeffreys). Astronomical Journal 71, No. 7, 1966, 568–578. 38) On the theory of quasiperiodic motions. SIAM Review, Vol. 8, No. 2, 1966, 145–172. 39) On the theory of quasiperiodic solutions of differential equations. Proc. Internat. Symposium, Mayaguez, P. R., 1965. Ed. by J. Hale and La Salle in Differential Equations and Dynamical Systems, Acad. Press, New York, 1967, 15–26. 40) On non-oscillating networks. Quarterly of Appl. Math., 25, 1967, 1–9. 41) Lectures on Hamiltonian Systems. Memoirs of the AMS, 61, 1968, 1–60. 42) Convergent series expansions for quasi-periodic motions. Math. Ann. 169, 1967, 136–176. 43) Quasi-periodic solutions in the three-body problem. Bull. Astronom. Ser. 3, Tome 3, 1968, 53–59. 44) Jointly with L. J. Laslett, E. M. McMillan. Long-Term Stability for Particle Orbits. AEC Research and Development Report NYO-1480-101, New York University, 1968, 49–58. 45) On a theorem of Anosov. Journal of Diff. Eq., Vol. 5, No. 3, 1969, 411–440. 46) On the boundedness of the solutions and the singularity of the St¨ormerproblem. Celestial Mechanics 2, 1970, 334–338. 47) Regularization of Kepler’s problem and the averaging method on a manifold. Comm. Pure Appl. Math., Vol. 23, 1970, 609–636.

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48) On the construction of almost periodic solutions for ordinary differential equations. Proc. Int. Conf. on Functional Analysis and related Topics, Tokyo, 1969, 60–67, University of Tokyo Press, 1970. 49) On pointwise estimates for parabolic differential equations. Comm. Pure Appl. Math., Vol. 24, 1971, 727–740. 50) A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. Journal, 20, 1971, 1077–1092. 51) On a nonlinear problem in differential geometry. Proceedings Symposium held at Univ. Bahia, Salvador, Aug. 1971 ed. M. Peixoto. Dynamical Systems, Academic Press, 1973, 273–280. 52) On a class of quasi-periodic solutions for Hamiltonian systems. Dynamical Systems. Proceedings Symposium held at Univ. of Bahia, Salvador, Aug. 1971, ed. M. Peixoto, Academic Press, 1973, 281–288. 53) Stability theory in celestial mechanics. Proc. Int. Astronomical Union, Warsaw 1975. The stability of the solar system and of small stellar systems, ed. Kozai 1974, 1–9. 54) A lemma on hyperbolic fixed points of diffeomorphisms. Uspekhi Mat. Nauk, 29, 2 (176), 1974, 228–232. 55) Neue Anwendungen klassischer Stabilit¨atsprobleme. 1. Vortrag: Ist das Sonnensystem stabil? Wolfgang Pauli-Vorlesungen ETH Z¨urich, WS 1974/75. (Publiziert in Neue Z¨urcher Zeitung, 14.5.1975). 56) Is the solar system stable? The Mathematical Intelligencer 1, No. 2, 1978, 65–71. English version of [53]. 57) Holomorphic equivalence and normal forms of hypersurfaces. Differential Geometry, part 2, Proc. Symposia in Pure Math., vol. 27, ed. Chern and Osserman, Am. Math. Soc. 1975, 109–112. 58) Jointly with S. S. Chern. Real hypersurfaces in complex manifolds. Acta Math. 133, 1974, 219–271. 59) Jointly with S. S. Chern, Real hypersurfaces in complex manifolds. Russian translation of (55.), Uspekhi Mat. Nauk, 38, 1983, 149–193. 60) Finitely many mass points on the line under the influence of an exponential potential — An integrable system. Proc. Battelle Rencontres Lecture Notes in Physics 38, 1975. 61) Three integrable Hamiltonian systems connected with iso-spectral deformations. Advances in Math. 16, 1975, 197–220. 62) Periodic orbits near an equilibrium and a theorem by a Alan Weinstein. Comm. Pure Appl. Math. 29, 1976, 727–747.

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63) A theorem by A. Weinstein and bifurcation theory. Report of the Univ. Louvain, January 1976. 64) The scattering problem for some particle systems on the line, Geometry and Topology. Lecture Notes in Math., Vol. 597, 1977, 441–463. 65) Proof of a generalized form of a fixed point theorem due to George D. Birkhoff, Geometry and Topology. Lecture Notes in Math., Vol. 597, 1977, 464–494. 66) Jointly with H. Airault and H. P. McKean. Rational and elliptic solutions of the Korteweg – de Vries equations and a related many body problem. Comm. Pure Appl. Math., Vol. 30, 1977, 98–148. 67) Jointly with M. Adler. On a class of polynomials connected with the Korteweg – de Vries equation. Comm. Math. Physics 61, 1978, 1–30. 68) On a class of polynomials connected with the Korteweg – de Vries equation. Proc. Uppsala 1977, Int. Conf. on Diff. Eq., Uppsala 1977, 144–154. 69) A fixed point theorem in symplectic geometry. Acta Math. 1978, 17–34. 70) Various aspects of integrable Hamiltonian systems. Proc. CIME Conf., Bressanone, 1978; also published in Progress of Mathematics, Vol. 8, Boston: Birkh¨auser, 1980, 233–289. 71) Various aspects of integrable Hamiltonian systems. Russian translation of [66], Uspekhi Mat. Nauk 36, 1981, 109–151. 72) Nearly Integrable Hamiltonian Systems, Am. Inst. Physics Conf. Proceedings 46, AIP 1978, 1–15. 73) Field Medals (III): A Broad Attack on Analysis Problems. Science, 202, 1978, 612–613. 74) The holomorphic equivalence of real hypersurfaces. Proc. Int. Congress of Mathematicians, Helsinki 1978, 659–668. 75) Stable and unstable motion in dynamical systems. Symposium in «Nonlinear orbit dynamics and the beam-beam interactions». AIP Conference Proceedings Nr. 57, Brookhaven, Nat. Lab. March 19–21, ed. M. Month, J. C. Herrera. American Institute of Physics, 1979, 222–235. 76) Hidden symmetries in dynamical systems. American Scientist, Vol. 67, Nr. 6, 1979, 689–695. 77) Geometry of quadrics and spectral theory. The CHERN Symposium 1979, Berkely, Springer Verlag New York, 1980, 147–187. 78) An example of a Schr¨odinger equation with almost periodic potential and nowhere dense spectrum. Comm. Math. Helvetici, 56, 1981, 198–224. 79) Jointly with R. Johnson. The rotation number for almost periodic potentials. Comm. Math. Phys. 84, 1982, 403–438.

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Curriculum Vitae

80) Integrable Hamiltonian Systems and Spectral Theory. Fermi Lectures, Pisa 1981. Lezioni Fermiane, Acad. Nat. dei Lincei, Pisa 1981. 81) Integrable Hamiltonian Systems and Spectral Theory. Revised, reprinted from [75] in the Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations, ed. Liao Shantao, S. S. Chern, Science Press, Beijing, China 1986, 157–229. 82) Jointly with S. Webster. Normal forms for real surfaces in C near complex tangents and hyperbolic surface transformations. Acta Math. 150, 1983, 255–296. 83) Jointly with S. Webster. Normal forms for real surfaces in C near complex tangents and hyperbolic surface transformations. Russian translation of [76], Uspekhi Mat. Nauk 41, 1986, 143–174. 84) Jointly with J. P¨oschel. On the stationary Schr¨odinger equation with a quasiperiodic potential. Physica 124A (1984), 535–542. 85) Jointly with J. P¨oschel. An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comm. Math. Helvetici, 59, 1984, 39–85. 86) Breakdown of Stability. 2 lectures held in Sardinia (CERN) Symposium). Lecture Notes in Physics No. 247, ed. Jowett, Month, Turner, Springer 1986, 492–518. 87) Analytic surfaces in C and their local hull of holomorphy. Annales Academiae Scientiarum Fenniae, Series A.I. Mathematica, Vol. 10, 1985, 397–410. 88) Monotone twist mappings and the calculus of variation. Ergodic Theory and Dynamical Systems 6, 1986, 401–413. 89) Recent developments in the theory of Hamiltonian systems. Expanded version of the John Neumann lecture held at the SIAM Conference in Seattle, July 1984. SIAM Review, 28, 1986, 459–485. 90) Minimal solutions of variational problems on a torus. H. Poincar´e: Analyse nonlin´eaire 3, 1986, 229–272.

Annales de l’Inst.

91) On the construction of invariant curves and Mather sets via a regularized variational principle. In Proc. Nato Advanced Research Workshop in Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P. Rabinowitz et. al. NATO ASI Series, Ser. C. Math. and Phys. Sciences vol. 209, Reidel Publ. Comp. 1987, 221–234. 92) Presidential Address at the 10th General Assembly of IMU, Oakland, Cal. USA, July 31 – Aug. 1, 1986. Bull. of the IMU, 26, 1986, 10–12. Addresses at the International Congress of Mathematicians 1986 Berkeley. Proceedings of the Int. Congress Math. 1986 in Berkeley, published by the AMS 1988. 93) Minimal foliations on a torus. Proceedings of the CIME Conference on Topics in Calculus of Variations, July 20–28, 1987 in Montecatini, Italy, Lecture Notes in Math. 1365, 1988.

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Curriculum Vitae

15

94) A stability theorem for minimal foliations on a torus. Ergodic Theory and Dynamical Systems 8, 1988, 251–281. ¨ 95) Uber die Stabilit¨atstheorie der Himmelsmechanik. Mitt. der Deutschen Akad. der Naturforscher Leopoldina (R.3) 33, 1987 (1989), 171–174. 96) Quasi-periodic solutions of nonlinear elliptic partial differential equations. Bol. Soc. Mat., Vol. 20, No. 1, 1989, 29–45. 97) Jointly with A. Veselov. Discrete version of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 1991, 217–243. 98) Jointly with B. Dacorogna. On a partial differential equation involving the Jacobian determinant. Annales de l’Inst. H. Poincar´e: Analyse nonlin´eaire 7, nr. 1, 1990, 1–26. 99) On commuting circle mappings and simultaneous Diophantine approximations. Mathematische Zeitschrift 205, 1990, 105–121. 100) Jointly with M. Struwe. On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus. Bol. Soc. Braz. Mat., Vol. 23, 1992, Ns. 1–2, 1–20. 101) On quadratic symplectic mappings. Mathematische Zeitschrift 216, 1994, 417–430. 102) An unusual variational problem connected with Mather’s theory for monotone twist mappings. Progess in Nonlinear Differential Equations and their Applications, Vol. 12. Seminar on Dynamical Systems, St. Petersburg 1991. Editors: Lazutkin, et al., Birkh¨auser Basel 1994, 81–89. 103) Smooth approximation of Mather sets of monotone twist mappings. Comm. Pure Appl. Math., Vol. 47, 1994, 625–652. 104) Remark on the smooth approximation of volume-preserving homeomorphisms by symplectic diffeomorphisms. Preprint FIM 1992. 105) Jointly with A. Veselov: Two dimensional «Discrete Hydrodynamics» and Monge – Amp`ere equation. Preprint FIM April 1993. 106) On the persistence of pseudo-holomorphic curves on an almost complex torus (with appendix by J¨urgen P¨oschel). Invent. math. 119, Springer-Verlag 1995, 401–442. 107) Laudatio f¨ur S. Hildebrandt, Bonn, gehalten anl¨asslich der von StaudtPreisverleihung, Erlangen 5.7.94. Mitteilungen der Deutschen Mathematiker Vereinigung 4, 1994, 6–12. 108) Pseudo-holomorphic curves on a torus. Proceedings of the Royal Irish Academy, Vol. 95A, Supplement, 13–21 (1995). 109) Ist das Sonnensystem stabil? Mitteilungen der Deutschen Mathematiker Vereinigung 4, 1996, 17–28.

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Curriculum Vitae

110) Jointly with Hans R. Jauslin and Heinz-Otto Kreiss: On the forced Burgers equation with periodic boundary conditions. Preprint FIM April 1997.

Books Jointly with C. L. Siegel: Lectures on Celestial Mechanics. SpringerVerlag, N.Y., 1971. Stable and Random Motions in Dynamical Systems, Princeton University, 1973. Integrable Hamiltonian Systems and Spectral Theory. Fermi Lectures, Pisa 1981. Lezioni Fermiane, Acad. Nat. dei Lincei, Pisa 1981. Edited works Dynamical Systems, Theory and Applications. Physics 38, Springer 1975.

Lectures Notes in

Collected Papers by Fritz John. 2 volumes, Birkh¨auser 1985.

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Editorial Note In 1999 the Russian publishing house Regular and Chaotic Dynamics released in Russian the first volume comprising works of a leading mathematician of the modern era Jurgen Moser. The author himself selected the works to be published. The idea of publishing a separate volume of J. Moser’s works occured during the preparation of publishing a special issue of the Regular and Chaotic Dynamics journal dedicated to the 70th birthday of J. Moser. J. Moser wished to exposed to the Russian reader to a broader range of his ideas, thus instead of a single volume we decided to publish three volumes. Each volume comprises works sharing the same idea and the same methods of solution. The second volume, KAM-theory and stability issues, was released in Russian in 2001. The third volume is scheduled to appear at the end of 2003. Based upon discussions with us, J. Moser decided the first volume ought to be devoted to integrable systems and some related issues. The second volume contains works on the KAM-theory, stability issues and invariant curves. The works collected in the third volume may be less well known to the Russian reader, they mostly deal with topology, averaging methods, normal forms and the theory of ordinary and partial differential equations. The works of each volume are arranged in chronological order to illustrate that throughout his life J. Moser revisited many of his ideas and methods, each time obtaining more profound and complete results. The style of the works of J. Moser is transparent. He never launches into lengthy discussions of vague generalities or uses confusing definitions and statements. He never makes the exposition too formal unless it is absolutely necessary. It should be noted that J. Moser inherited this style of writing from his teacher C. Zigel. We hope that the three-volume collection of J. Moser’s works in English will be useful for a broad range of mathematicians, engineers and physicists. For a young researcher his works serve as an example illustrating all stages of scientific exploration: from the formulation of a problem to a lucid presentation of the results. This edition is issued in commemoration of Jurgen Moser who died on 17 Dec 1999 at age 71, in Zurich.

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Finitely Many Mass Points on the Line under the Influence of an Exponential Potential. — An Integrable System1

§ 1. Analogue of the Toda Lattice for Finitely Many Mass Points We consider the analogue of the Toda lattice [8] where only a finite number of mass points are admitted which move freely on the real axis. Denoting the position of the mass points by xk , k = 1, . . . , n, we form the Hamiltonian n

n−1

k=1

k=1

  yk2 + e(xk −xk+1 ) H= 1 2

(1.1)

with the differential equations x˙ k = Hyk = yk , y˙ k = −Hxk = e

k = 1, 2, . . . , n, xk−1 −xk

y˙ 1 = −Hx1 = −e

x1 −x2

,

− exk −xk+1 ,

k = 2, 3, . . . , n − 1,

(1.2)

y˙ n = −Hxn = exn−1 −xn . Thus we can write our system (1.2) as xk = exk−1 −xk − exk −xk+1 ,

k = 1, . . . , n.

(1.2′ )

if we set ex0 −x1 = 0 and exn −xn+1 = 0, that is we have the formal boundary condition x0 = −∞, xn+1 = +∞. (1.3) It is the aim to study completely the flow determined by this system of differential equations and relate the solution to the existence of n integrals of the 1 Proc.

Battelle Rencontres Lecture Notes in Physics, 38, 1975.

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Finitely Many Mass Points on the Line

motion. These integrals are essentially the same as those found by Henon [4] and Flaschka [1] for the same system of differential equations (1.2′ ) under periodic boundary conditions, say xk+n = xk + 1,

yk+n = yk ,

k = 0, ±1, . . .

(1.3′ )

The crucial difference between the two problems is that the boundary condition (1.3′ ) gives rise to a compact energy surface and the solutions are expected to be quasiperiodic, lying on tori, as one is familiar from integrable Hamiltonian systems. If we impose the boundary condition (1.3) instead of (1.3′) the energy surface is noncompact, as the particles can run to infinity. In fact, we will show, as is intuitively clear, that for any initial configuration mutual distances between all particles grow indefinitely, i. e. xk−1 − xk → ∞ for k = 2, . . . , n; and they behave asymptotically like free particles depending linearly on time. This suggests the scattering problem: To determine the relation between this asymptotic motion for the past and the future. This can be done explicitly here and one finds that yn−k+1 (+∞) = yk (−∞), so that at t = +∞ the first particle has the velocity of the last at t = −∞ etc. as in a familiar experiment of collision of steel balls. Moreover, the phase relation can also be determined explicitly and we will show that   log(yj− − yk− )2 , log(yj− − yk− )2 − xn−k+1 (t) − xk (−t) − 2yk− t → jk

yj−

= yj (−∞) are assumed ordered according to size. Thus the particles where behave asymptotically as if they interacted just pairwise! This will be derived in Section 4. In the limit t → +∞, the yk , k = 1, 2, . . . , n, or their symmetric functions, are t-independent integrals of the motion, and one may ask for integrals of the given system which asymptotically agree with these integrals. This is indeed possible, and Henon’s construction of integrals was based on this idea, even though in the periodic case this idea is not really justified and was only a guiding principle for the construction of integrals. For the noncompact case, i. e. boundary condition (1.3), the free system is indeed the limit state and this approach quite natural. On the other hand, the noncompact case is, of course, much less complicated, as the solutions have no recurrence property and the flow has the nature of parallel flow. In fact, we will show that (1.2) can be mapped into the following system of differential equations, dλk = 0, dt

drk = − ∂V , dt ∂rk

k = 1, . . . , n,

(1.4)

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§ 2. Flaschka’s Form of the Differential Equation and Asymptotic Behavior

21

where V =

n 

λk rk2

k=1 n 

2

k=1

(1.4′ ) rk2

and the variables are restricted to the (2n − 1) dimensional domain λ1 < λ2 < . . . < λn ;

n 

rk2 = 1,

rk > 0.

(1.5)

k=1

Clearly, the solutions run from the maximum of V at rk = δkn to the minimum of V of rk = δk1 as t runs from −∞ to ∞, and λ1 , λ2 , . . . , λn are integrals of the motion, while r1 , . . . , rn−1 can be viewed as parameters on the surfaces λk = const. The mapping taking x, y into the variables λk , rk on (1.5) is up to translation of the xk one to one and will be given explicitly. The inverse mapping illustrates the inverse method of spectral theory. Thus this note does not claim any new idea and should be considered as providing a simple model illustrating the construction of integrals and its connection with the inverse method of spectral theory in extreme simplicity, yet with all rigor. On the other hand, it leads immediately to an unsolved problem if one wants to carry out this approach for the periodic boundary condition (1.3′ ). Although the integrals Ik for this problem are well known, no parameters are known on the level surfaces Ik = ck which determine the xk (mod 1), yk uniquely. This is related to the lack of an inverse theory for the Hill’s equation −u′′ + q(x)u = = λu, q(x + 1) = q(x) under periodic boundary conditions u(x + 1) = u(x) where the problem consists in finding a set of quantities which together with the eigenvalues allow one to determine q(x). One can hope to shed some light on this question if one could solve the above finite dimensional problem.

§ 2. Flaschka’s Form of the Differential Equation and Asymptotic Behavior We set, with Flaschka, ak = 1 e(xk −xk+1 )/2 , 2

bk = − 1 yk 2

(2.1)

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Finitely Many Mass Points on the Line

so that the differential equations (1.2) go into a˙ k = ak (bk+1 − bk ), b˙ k = 2(a2k − a2k−1 ),

k = 1, 2, . . . , n − 1

k = 1, 2, . . . , n

(2.2)

with the boundary conditions (1.3) being a0 = 0,

an = 0.

(2.3)

Observe that (2.1) provides a transformation of the (x, y) variables into the (a, b)-variables. We identify points (x, y), ( x, y) if xk − x k is independent of k, and call the equivalence class a «configuration». It is characterized by the (2n − 1) numbers xk − xn , k = 1, . . . , n − 1, and yk , k = 1, . . . , n. Thus (2.1) defines an invertible transformation of the (2n − 1)-dimensional space of configurations into the domain   D = a, b | ak > 0, k = 1, . . . , n − 1 , and it remains to study the flow given by the quadratic differential equation (2.2) in D. The energy is given by H=4

n−1  k=1

 n  b2k . a2k + 1 2

(2.4)

k=1

We show first that for any solution in D ak (t) → 0

for t → ±∞ and k = 1, . . . , n − 1,

(2.5)

which amounts to the assertion that xk+1 − xk → ∞ as t → ±∞. To prove this we consider the system (2.2) with prescribed a0 (t), an (t) ∈ L2 (−∞, +∞) so +∞ that (a20 + a2n ) dt < ∞ and prove the −∞

Lemma. For any solution of (2.2) with this modified boundary condition we have ∞ (a21 + a2k−1 ) dt < ∞. −∞

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§ 2. Flaschka’s Form of the Differential Equation and Asymptotic Behavior

23

PROOF. Consider the function ϕ(t) = b1 − bn

for which

dϕ = b˙ 1 − b˙ n = 2(a20 + a2n ) − 2(a21 + a2n−1 ). dt

Thus ψ = 1ϕ − 2

t

(a20 + a2n ) dt

−∞

satisfies

dψ = −(a21 + a2n−1 ). dt Since by the energy relation ϕ and hence ψ is bounded, also T

(a21 + a2n−1 ) dt = ψ(−T ) − ψ(T )

−T

is bounded for T → ±∞, proving the lemma. We can apply this argument, in particular, to a0 = an = 0. Applying this lemma to the reduced system where the first and last equations in the first and +∞ second line of (2.2) are cancelled we conclude that (a2n + a2n−2 ) dt < ∞ and −∞

inductively that +∞ n−1  k=1−∞

a2k dt < ∞.

(2.6)

 Since on the other hand |p| ˙  2 a2k |bk − bk+1 |  M is bounded it follows n−1  2 that p = ak → 0 for t → ±∞. Indeed, otherwise there would exist a se1

quence tk → ∞ with p(tk )  δ > 0. We may assume that the sequence is so selected that |tk+1 − tk |  δ/M . Since p(t)  δ/2 in the disjoint intervals |t − tk | < 1 δ it cannot be integrable, contradicting (2.6). This 2M

proves (2.5). Moreover, we conclude from (2.2) that bk tends to a limit bk (∞) as t → +∞.

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Finitely Many Mass Points on the Line

Flaschka [1, 2] noted that the above system (2.2) can be expressed in matrix form

d L = BL − LB dt

where ⎛

⎞ 0 b 1 a1 ⎟ ⎜ a1 b 2 ⎟ ⎜ ⎟ ⎜ . .. L=⎜ ⎟; ⎜ ⎟ ⎝ bn−1 an−1 ⎠ 0 an−1 bn

(2.7)

⎞ 0 0 a1 ⎟ ⎜−a1 0 ⎟ ⎜ ⎟ ⎜ . .. B=⎜ ⎟. ⎟ ⎜ ⎝ 0 an−1 ⎠ 0 −an−1 0 ⎛

Thus if U = U (t) is the orthogonal matrix satisfying dU = BU ; dt

U (0) = I

then by (2.7) d (U −1 LU ) = 0 dt hence U −1 LU = L(0). Thus, L(t) is similar to L(0) and the eigenvalues λk of the Jacobi matrix L, which are real and distinct, are independent of t. This description of the integrals as eigenvalues of a linear operator is due to Lax [5] and Flaschka’s derivation was based on his approach. Thus the characteristic polynomial ∆n (λ) = det(λI − L) =

n 

k=1

(λ − λk ) =

n 

Ik λn−k

(2.8)

k=0

as well as the coefficients I1 , . . . , In are constants of the motion (2.2). For definiteness we order the eigenvalues according to their size, λ1 < λ2 < . . . < λn . Notice that L(t) → L(∞) as t → +∞ where L(∞) is a diagonal matrix whose diagonal elements must be the eigenvalues λk in appropriate order. From a˙ k ak ∼ bk+1 (∞) − bk (∞)

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§ 3. Partial Fractions and Continued Fractions

25

and (2.5) we conclude that bk+1 (∞) < bk (∞) or bk (∞) = λn−k+1 i. e. L(∞) = diag(λn , λn−1 , . . . , λ1 ). Using that the t-reversing substitution t → −t;

ak → an−k ;

bk → bn+1−k

leaves the system invariant, we conclude that L(−∞) = diag(λ1 , λ2 , . . . , λn ) i. e. L(∞), L(−∞) differ just in the order of the diagonal elements. The physical interpretation of this result is: If for t → −∞ the particles xk approach the velocities yk = −2λk where y1 < y2 < . . . < yn then for t → +∞ the particles xk have the velocities yn−k+1 so that the particles exchange their velocities. This describes the flow for our problem (2.2), (2.3). Still we will find another set of variables, rk > 0, k = 1, . . . , n − 1, which together with the λk form a set of coordinates, and represent the differential equations in these new variables.

§ 3. Partial Fractions and Continued Fractions Let R(λ) = (λI − L)−1 where we suppress the dependence in t. This is an n by n matrix and we single out the element in the last row and last column Rnn (λ) = (R(λ)en , en ) = f (λ)

where en = (0, 0, . . . , 0, 1),

(3.1)

and f (λ) is hereby defined. Since L is symmetric it follows that f (λ) is an analytic function for Im λ = 0 and Im f (λ) > 0

for

Im λ > 0.

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Finitely Many Mass Points on the Line

Moreover, it is rational with simple poles at the eigenvalues λk and so admits the partial fraction expansion f (λ) =

n 

k=1

rk2 , λ − λk

rk > 0,

(3.2)

with positive residua rk2 . Moreover, for |λ| → ∞ one has λf (λ) → 1 and n 

rk2 = 1.

k=1

Thus we have a mapping ϕ associating with every point in D = {a1 , . . . , an−1 , b1 , . . . , bn with ak > 0}

(3.3)

a point in Λ = {λ1 , . . . , λn , r1 , . . . , rn with λ1 < λ2 < . . . < λn , n  rk2 = 1, rk > 0}.

(3.4)

k=1

We claim that this mapping ϕ : D → Λ is one to one and onto. We will view it as a coordinate transformation and then describe the differential equations in the new variables. The fact that the mapping ϕ has an inverse ϕ−1 : Λ → D corresponds to the inverse method of spectral theory, which in the elementary form described here goes back to Stieltjes [3]. It is based on the fact that f (λ) admits a continued fraction expansion f (λ) =

1 a2n−1 λ − bn − λ − bn−1 − . .

(3.5) 2 . − a1

λ − b1

where the entries ak , bk agree precisely with those of L. To prove this we establish the identity f (λ) =

∆n−1 ∆n

(3.6)

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§ 3. Partial Fractions and Continued Fractions

27

where ∆n is the characteristic polynomial of (λI − L), see (2.8), and ∆k the k by k subdeterminant obtained by canceling the last n− k rows and columns of (λI − L). Expanding ∆k by the last row one finds ∆k = (λ − bk )∆k−1 − a2k−1 ∆k−2

(3.7)

for k = 3, 4, . . . , n; it holds also for k = 1, 2 if we set ∆−1 = 0,

∆0 = 1.

Thus the ratios sk = ∆k /∆k−1 satisfy the recursion formula a2k−1 sk = λ − b k − s for k = 2, 3, . . . , n k−1 which leads to a finite continued fraction for sn =

∆n = f −1 (λ). ∆n−1

Thus the representation (3.5) follows from (3.6) which we prove now. For this purpose we compute the last column R en = z

of R = R(λ).

We find ⎧ ∆k ⎪ ⎪ ⎪ ⎨ zk+1 = ∆n ak+1 . . . an−1 for k = 0, 1, . . . , n − 2, ⎪ ∆n−1 ⎪ ⎪ . ⎩ zn = ∆n

(3.8)

Indeed z is the solution of

(λI − L)z = en and using the recursion formula one readily verifies (3.8). Thus f (λ) = Rnn = zn =

∆n−1 ∆n

as we wanted to show. Thus for a given matrix L we can compute the rational function f (λ) which has n simple real poles with positive residua, since Im f (λ) > 0 for Im λ > 0.

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Finitely Many Mass Points on the Line

Ordering these poles according to size we have defined the mapping ϕ taking D into Λ (see (3.3), (3.4)). We come to the «inverse problem» which requires that we determine ϕ−1 . With any point in Λ we associate f (λ) by (3.2). Then Im f > 0 for Im λ > 0 and λf (Λ) → 1 for |λ| → ∞. Thus 1 = λ + A − g(λ) f (λ) where A is a real constant and g(λ) is a rational function which satisfies Im g(λ) = Im λ +

Im f > 0 for Im λ > 0. |f |2

Thus g(λ) has only simple poles on the real axis and their number is n − 1. One 2  2 2   > 0. Thus computes easily −A = λk rk2 , λg(λ) → λk rk − λk rk2 1

g = Bfn−1 with B > 0, and λfn−1 → 1 for |λ| → ∞. Thus f (λ) =

1 λ + A − Bfn−1

and by induction we get a unique continued fraction of the form (3.5) with A = −bn , B = a2n−1 > 0, etc. This shows that ϕ maps D one to one onto Λ. Finally we express the differential equation (2.2) in these new variables. For this purpose we deduce from (2.7) d R = R dL R = BR − RB dt dt and taking the last element Rnn = f in R we find df = (en , (BR − RB)en ) = −2(en , Ran−1 en−1 ) = −2an−1 Rn, n−1 . dt Since Rn, n−1 agrees with zn−1 in (3.8) we obtain ∆n−2 df = −2a2n−1 . ∆n dt

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§ 3. Partial Fractions and Continued Fractions

29

This formula allows us to determine the desired differential equations. Since we established already that dλk /dt = 0 we have n

 2rk r˙k df = . dt λ − λk k=1

Comparing the residue of the last two expressions we get  ∆n−2  2 2rk r˙k = −2an−1 .  ∆′n  λ=λk

By the recursion formula (3.7) we have

∆n = (λ − bn )∆n−1 − a2n−1 ∆n−2 or, since ∆n (λk ) = 0, ∆n−2 (λk ) = hence

λk − bn ∆n−1 (λk ), a2n−1

 ∆n−1  2rk r˙k = −2(λk − bn )  ∆′n 

. λ=λk

A similar comparison of the residua of

n

f (λ) =

 rk2 ∆n−1 = ∆n λ − λk k=1

gives

 ∆n−1   ∆′n 

hence

Since

n 

k=1

= rk2 λ=λk

2rk r˙k = −2(λk − bn )rk2 . rk2 = 1 we find 0=



rk r˙k = −



λk rk2 + bn

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30 and so, bn =

Finitely Many Mass Points on the Line



λk rk2 , as we had seen before, and    r˙k = − λk − λj rj2 rk

which gives the differential equation (1.4) of Section 1. These differential equations represent the vector field along the gradient of the function V (r) (see (1.4′ )) restricted to the part of the unit sphere lying in the positive quadrant. Thus every solution approaches for t → +∞ the minimum: r(t) → e1 and for t → −∞ the maximum: r(t) → en . Of course, it is also possible to give an analytical representation for the solutions, since they are obtained by projecting the linear differential equations r˙k = −λk rk on the unit sphere. Thus we find ⎧ λk (t) = λk (0), ⎪ ⎪ ⎪ ⎪ ⎨ r2 (0)e−2λk t . rk2 (t) = n k ⎪  2 ⎪ −2λj t ⎪ rj (0)e ⎪ ⎩ j=1 To summarize our result we consider the rk as homogeneous variables, still positive, and set, accordingly,  −1  n n rk2 f (λ) = . (3.9) rk2 λ − λk k=1

k=1

From (3.5) and the calculation of continued fractions it is clear that the a2k , bk are rational functions of rj , λj of degree 0 in the rj . One verifies that ak , bk are of degree 1 in λj . Thus we have the following rational transformation a2k = Ak (r, λ), b2k

= Bk (r, λ),

k = 1, 2, . . . , n − 1, k = 1, 2, . . . , n

(3.10)

of λ1 < . . . < λn ; rk > 0 into the domain D. If we identify two proportional vectors r the mapping is one to one. In these homogeneous coordinates rk the differential equations become linear drk dλk = 0; = −λk rk . (3.11) dt dt Thus the solutions of (2.2) can be represented as rational functions of the n constants λj and n exponential functions e−λj t .

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31

As we mentioned in Section 1 the flow is particuarly simple in this case since no periodic or recurrent solutions are present. In the more interesting case of the periodic boundary condition one has quasiperiodic solutions and the task of finding coordinates for the integral surfaces which are tori, is more difficult. The main problem is the «inverse problem» which consists in recovering L — which is then a cyclic matrix — from its eigenvalues and appropriately chosen quantities. This problem seems unsolved as yet.

§ 4. Solution of the Scattering Problem From the results of Section 2 it follows that the asymptotic behavior of the solutions of our problem (1.2′ ) is given by + −δt xk (t) = α+ ), k t + βk + o(e − −δt ) xk (−t) = −α− k t + βk + o(e

(4.1)

for t → +∞ with some δ > 0. Moreover, we found α+ k =

lim

k→k+∞

yk = −2λn−k+1

and α− k = −2λk

i. e. − α+ n−k+1 = αk

(4.2)

which expresses that the (n − k + 1)st particle has then for t → +∞ velocity which the k th particle had in the past. Our goal is to determine the relation between the phases βk+ , βk− which can be given explicitly too. This remarkable fact is also a consequence of the integrable character of the system and the representation of exk −xk+1 , yk as rational functions of λj , e−λj t given by (3.10), (3.11). An explicit calculation seems prohibitive; nevertheless the following argument, which uses just rudimentary properties of rational functions will lead to the goal. The result is  + ϕjk (α− ) (4.3) = βk− + βn−k+1 j =k

where ϕjk (α) =



− 2 log(α− j − αk )

− 2 − log(α− j − αk )

for j < k, for j > k.

(4.4)

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Finitely Many Mass Points on the Line

For n = 2 this amounts to − 2 β2+ = β1− − log(α− 1 − α2 ) ,

− 2 β1+ = β2− + log(α− 1 − α2 ) .

(4.5)

− Thus ϕjk represents the phase shift between two particles with velocities α− j , αk at t = −∞. The result (4.3) can therefore be interpreted as follows: The particles are scattered just as if their interaction takes place two at a time! This was suggested to me by M. Kruskal who described an analogous phenomenon for solutions of the Korteweg – de Vries equation (see [6], Theorem 3.7) and by P. D. Lax. This phenomenon which had been discovered by Zakharov et al. (see [6] for references) is obviously intimately related to our result and it is conceivable that one can be derived from the other — but we have not pursued this point. We illustrate the statement in Figures 1, 2. Figure 1 illustrates the case n = 2, which is given explicitly in terms of cosh(λ2 − λ1 )t. The asymptotic behavior can be interpreted as the elastic reflection of two rods of length − 2 ϕ21 = log(α− 2 − α1 ) , provided this number is positive. For negative values of ϕ21 the particles reflect only after passing each other. However, this interpretation is somewhat misleading, especially if n > 2, since the length of the rods depends on their velocity, not on the label. We indicate schematically the construction of the scattering for n = 3 in Figure 2.

Fig. 1

Fig. 2

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33

To prove (4.3) first translate it into an asymptotic statement for (2.2). For this purpose we note that on account of the linear t-dependence of the center of mass we have n n   + βk = βk− k=1

k=1

− − βk− , i. e. it and therefore it suffices to prove (4.3) for the differences βk+1 suffices to establish   + + − − βn−k+1 = βk+1 − βk− − ϕjk + ϕj, k+1 . βn−k j =k

j =k+1

Using (2.1), (4.1) this amounts to lim an−k (t)ak (−t)e2(λk+1 −λk )t = Ck (λ),

t→+∞

with log(4Ck )2 = −



ϕj, k +

j =k

k = 1, 2, . . . , n − 1



ϕj, k+1 .

j =k+1

Finally, with α− k = −2λk and (4.4) this gives for Ck the expression   (λj − λk ) (λk+1 − λj ) j>k

Ck = 

j 1, λ3 − λ2

→ B D

for

λ2 − λ1 < 1. λ3 − λ2

but

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Finitely Many Mass Points on the Line

− where Φk (α) is a real analytic function of α = (α− 1 , . . . , αn ), independent − of βj . This follows from the fact that Ck (λ) is independent of rj (0), and

depends on λj = − 1 α− j only. For n + 1 particles we denote the corresponding 2

− α), α  = (α− function by ψk ( 1 , . . . , αn+1 ). The induction proof requires the verification of

ψk ( α) = Φk (α) + ϕn+1, k

for k = 1, 2, . . . , n.

(4.14)

The determination of ψn+1 ( α) follows then from n+1 

ψk ( α) = 0

k=1

which is a consequence of the linear t-dependence of the center of mass. Thus it suffices to prove (4.14). Since both sides are independent of the βj− we may, − and will, choose βn+1 very large positive, so that the n particles x1 , x2 , . . . , xn have already undergone their mutual interaction and are very far apart by the time xn+1 interacts with any of them. In other words, when xn+1 (t) exerts some force on x1 , . . . , xn there are already close to + xk ∼ α+ k t + βk ,

− where βk+ = βn−k+1 + Φn−k+1

and t is large positive. Thus the interaction of xn+1 with xk (t), k  n takes essentially place pairwise. Since before the interaction with xn+1 we have − xn−k+1 ∼ α− k t + βk + Φ k ,

k = 1, 2, . . . , n,

we obtain after interaction − xn−k+1 ∼ α− k t + βk + Φk + ϕn+1, k − which shows that ψk ∼ Φk + ϕn+1, k if βn+1 → ∞. But since ψk , Φk are − independent of β the assertions (4.14) follow. The situation is depicted in Figure 3.

§ 5. Associated Differential Equations The above Hamiltonian system (1.2) possesses, according to the above, n integrals I1 , I2 , . . . , In , which incidentally are polynomials in yk

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§ 5. Associated Differential Equations

39

Fig. 3

and exk −xk+1 . One may show, which we will not do here, that these integrals are in involution, i. e. the Poisson bracket for any two of these vanishes. Using the integrals as new Hamiltonians one can introduce n new vector fields which possess the same integrals and commute with each other. This makes the manifolds Ik = const into commutative groups. These are well known facts for integrable Hamiltonian systems (see, for example, Appendix 26 of [7]) which we will verify here directly. As a starting point we take the differential equation (2.7) which represents a deformation of the Jacobi matrix L leaving the spectrum fixed. But there are many such isospectral deformations corresponding to different choices of B. We restrict ourselves to skew symmetric matrices B giving rise to orthogonal similarity transformations. But instead of permitting only one pair of off diagonals we allow several. Let Bp stand for a skew symmetric matrix with p off diagonals above and adjacent to the diagonal. Thus the matrix B defined below (2.7) would be denoted by B1 . We claim that for every p in 1  p < n we can find nontrivial matrices Bp such that L˙ = Bp L − LBp defines a meaningful differential equation, that is that the commutator Bp L − LBp has only one off diagonal above the diagonal, while the others all vanish. We will establish this assertion below but point out first that all these differential equations have the eigenvalues

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Finitely Many Mass Points on the Line

of L as integrals and can therefore be transformed into the variables rk , λk as λ˙ k = 0, r˙k = fk (λ, r). For p = 2 one finds the matrix ⎛

0

β1

⎜ ⎜−β1 0 ⎜ B2 = ⎜ ⎜−γ1 −β2 ⎜ .. ⎝ .

where

βk = (bk + bk+1 )ak , γk = ak ak+1 ,

γ1 β2 0

..

.

..

.

−γn−2 −βn−1

⎞

⎟ ⎟ ⎟ γn−2 ⎟ ⎟ ⎟ βn−1 ⎠ 0

k = 1, 2, . . . , n − 1, k = 1, 2, . . . , n − 2

(5.1)

and the differential equation L˙ = B2 L − LB2 takes the explicit form a˙ k = ak (a2k+1 − a2k−1 + b2k+1 − b2k ), b˙ k = 2bk (a2k − a2k−1 ) + 2bk+1 a2k − bk−1 a2k−1 ,

k  n − 1, k  n,

(5.2)

where we set a0 = 0, an = 0. Introducing r, λ again by the transformation (3.10) we find the differential equation dλk = 0, dt

drk = −λ2k rk . dt

We just indicate the calculation. First restricting rk to

On the other hand

 2rk r˙k df = . dt λ − λk

(5.3) 

rk2 = 1 we have

df ˙ = (R(λ)e n , en ) = ((B2 R − RB2 )en , en ) = dt = −2(RB2 en , en ) = −2(R(γn−2 en−2 + βn−1 en−1 ), en ) = = −2(γn−2 Rn, n−2 + βn−1 Rn, n−1 ). With (5.1) and Rn, n−1 =

∆n−2 an−1 , ∆n

Rn, n−2 =

∆n−3 an−2 an−1 ∆n

(5.4)

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§ 5. Associated Differential Equations

we get 2

an−1 2 df (a ∆n−3 + (bn−1 + bn )∆n−2 ). = −2 dt ∆n n−2 Using the recursion formulae (3.7) for k = n − 1, n we find ∆n−1 df . = −2(λ2 − b2n − a2n−1 ) ∆n dt Comparing the residue of these expressions at λk with those of (5.3) we find r˙k = −(λ2k − b2n − a2n−1 )rk , or using rk as homogeneous coordinates r˙k = −λ2k rk . Thus the solutions of (5.2) can be expressed as rational functions of λk 2 and e−λk t and the asymptotic behavior of its solutions is also completely understood from the results of the previous sections. It is interesting to observe that the differential equations (5.2) possess bk = = 0, k = 1, . . . , n, as an invariant manifold on which they reduce to a˙ k = ak (a2k+1 − a2k−1 ),

k = 1, . . . , n − 1

(5.4)

which are the deformation equations for a Jacobi matrix L with a zero diagonal1. To understand which of the solutions of (5.3) corresponds to (5.4) we consider again the continued fraction expansion (3.5), denoting the left-hand side by f (λ, a, b). One easily verifies that the involution bk → −bk , ak → ak gives rise to n  rk2 −f (−λ, a, −b) = f (λ, a, b) = . λ − λk k=1

Hence, since the eigenvalues λk are ordered according to size, the above involution corresponds to λk → −λn−k+1 ,

rk → rn−k+1 .

1 These equations for n = ∞ were recently studied by M. Kac and van Moerbeke, according to a letter from M. Kac.

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The fixed points of this involution are the points (a, b) with bk = 0 in the first representation and the points (λ, r) with λk + λn−k+1 = 0,

rk → rn−k+1 .

(5.5)

This is evident also from the fact that the symmetric Jacobi matrices with zero diagonal have a spectrum symmetric with respect to the origin. Thus the solutions 2 of (5.4) are given by precisely those rational functions in λj , e−λj t for which the λj satisfy (5.5). Using (2.1) it is easy to rewrite the system (5.2) in the variables xk , yk and one finds a Hamiltonian system x˙ k =

∂H2 , ∂yk

with H2 = − 1 6

n 

k=1

y˙ k = −

∂H2 , ∂xk

k = 1, 2, . . . , n,

n

 yk3 − 1 yk (exk−1 −xk + exk −xk+1 ). 2 k=1

Again, in the above system one has to set x0 = −∞, xn+1 = +∞. Although this system has no physical interpretation one has a full description of the scattering problem. If one expresses the above Hamiltonian H2 in terms of a, b one finds readily n

 λ3k . H2 = 4 tr L3 = 4 3 3 k=1

Since our original Hamiltonian (1.1) is given by H = 2 tr L2 = 2

n 

λ2k ,

k=1

one can expect that the further differential, equations are associated with Hamiltonian proportional to tr(Lp+1 ). We will not follow this up but conclude with establishing the existence of the matrices Bp for p = 1, 2, . . . , n − 1. It is convenient to write the matrices as difference operators. Let ξ stand for a double infinite sequence with components ξk (k integers) and let σ denote the shift operator (σ ξ)k = ξk+1 .

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§ 5. Associated Differential Equations

43

We will assume that ξk = 0 if k  0 or k > n and write the matrix L in the form L ξ = a(σ ξ) + bξ + σ −1 (a ξ). Here a, b stand for sequences with components ak and (a ξ)k = ak ξk . Thus σ(a ξ) = σ(a) · σ(ξ). In this notation Bp will be presented by Bp ξ = γσ p ξ + . . . + β(σ q ξ) + . . . − σ −q (β ξ) + . . . − σ −p (γ ξ),

(5.6)

where the q th order term indicates a typical term, 1  q < p. The commutator [Bp , L] = Bp L − LBp contains σ, σ −1 to powers up to p + 1. In fact, the highest order terms of this commutator are given by [Bp , L] = {γσ p (a) − aσ(γ)}σ p+1 + . . . and we determine the γ1 , γ2 , . . . , γn−1 so that (γσ p (a) − aσ(γ))ξ = 0

(5.7)

i. e. γk ak+p − ak γk+1 = 0,

k = 1, 2, . . . , n − p − 1.

This can be satisfied by γk = ak ak+1 . . . ak+p−1 ,

k = 1, 2, . . . , n − p.

Now we proceed inductively and determine the coefficients β of σ q in (5.6) to remove the terms of order q + 1 in [Bp , L], decreasing q from q = p to q = 1. Analogously to (5.7) this gives an equation of the form (βσ q (a) − aσ(β))ξ = g · ξ where g is a given sequence. In components βk ak+q − βk+1 ak = gk ,

k = 1, 2, . . . , n − q − 1.

These are n − q − 1 equations for n − q unknowns. The solution is therefore not unique, but if β1 is fixed arbitrarily, these equations can be solved recursively and uniquely, since ak > 0.

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Thus Bp can be so determined that in [Bp , L] all coefficients of σ q+1 for q = 1, 2, . . . , p vanish. Since [Bp , L] is symmetric it is a Jacobi matrix, giving rise to the desired differential equation L˙ = [Bp , L].

(5.8)

These are clearly the analogues of the higher order Korteweg – de Vries equations. Finally, it is obvious that the multiplication (r ⊗ r)k = rk rk

introduces a group structure into the manifolds λk = const making the (n − 1) dimensional manifold of Jacobi matrices L with fixed spectrum into an Abelian group. This group action commutes with the vector field (2.2), and more generally with the vector fields (5.8) for p = 1, 2, . . . , n − 1.

References [1] H. Flaschka, The Toda Lattice, I, Phys. Rev. B 9, (1974) 1924–1925. [2] H. Flaschka, The Toda Lattice, II, Prog. of Theor. Phys. 51 (1974) 703–716. [3] F. R. Gantmacher, and M. G. Krein, Oszillationsmatrizen, Oszillationskerne und Kleine Schwingungen Mechanischer Systeme, Akad. Verlag, Berlin (1960). See, Anhang II. [4] M. Henon, to appear in Phys. Rev. B 9, (1974) 1921–1923. [5] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), pp. 467–490. [6] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Korteweg – de Vries Equation and Generalizations VI, Methods for Exact Solutions, Comm. Pure Appl. Math. 27 (1974) 97–133. [7] V. I. Arnold, and A. Avez, Probl`emes Ergodiques de la M´ecanique Classique, Gauthiers-Villars, Paris (1967). [8] M. Toda, Wave propagation in anharmonic lattices, Jour. Phys. Soc. Japan 23 (1967) 501–506.

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Three Integrable Hamiltonian Systems Connected with Isospectral Deformations1 DEDICATED TO STAN ULAM

§ 1. Introduction (a) Background. In the early stages of classical mechanics it was the ultimate goal to integrate the differential equations of motions explicitly or by quadrature. This led to the discovery of various «integrable» systems, such as Euler’s two fixed center problems, Jacobi’s integration of the geodesics on a three-axial ellipsoid, S. Kovalevski’s motion of the top under gravity for special ratios of the principal moments of inertia, to name a few nontrivial examples. These efforts and their climax with the work of Jacobi who applied skillfully the method of separation of variables to partial differential equations, the Hamilton – Jacobi equations associated with the mechanical system, to establish their integrable character. However, this development took a sharp turn when Poincar´e showed that most Hamiltonian systems are not integrable and gave arguments indicating the nonintegrability of the three-body problem. In the same negative direction lies Brun’s discovery that the three-body problem has no algebraic integral except for the well-known classical ones and algebraic functions of these. These results express, in other words, that integrability of Hamiltonian systems is not a generic property; it is destroyed under small perturbations of the Hamiltonian. Therefore it seems an anachronismus to discuss these exceptional integrable systems nowadays. However, in recent years various phenomena were discovered which are clearly intimately related to integrable Hamiltonian systems yet they have very different origin. One is related to the discovery by Kruskal and others [6] of so-called solitons for the Korteweg – de Vries equation. These are wave solutions of a nonlinear partial differential equation having a strong stability behavior. Originally these phenomena were brought to light by numerical 1 Adv.

Math., v. 16 (1975) 197–220.

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Three Integrable Hamiltonian Systems

experiments and later on related to the existence of infinitely many conservation laws that restrict the evolution of the solutions severely. If one interprets the partial differential equation, in this case the Korteweg – de Vries equation, as a Hamiltonian system in an infinite-dimensional function space, with a certain symplectic structure, and the conservation laws as integrals of this system, one can view this as an example of an integrable system of infinitely many degrees of freedom. This was made precise in the work of Zakharov and Faddeev [15]. In an entirely unrelated development Calogero [2, 3] found that the quantum theoretical problem of n mass points on the line interacting under the influence of a potential proportional to the inverse square of the distance can be solved explicitly, and he conjectured that the corresponding classical problem might be integrable. This was established by Marchioro for the «three-body problem» by explicit calculation. Moreover, Calogero used his formula to study the scattering problem associated with the n-particle system in the quantum theoretical framework and found that the scattering is essentially trivial, in the sense that the particles behave asymptotically like elastically reflected mass points. (b) Results. It is our goal to show a close algebraic connection between these so different problems. However, instead of studying the infinite dimensional problems related to the partial differential equation in the one and the quantum theoretical framework in the other case, we will restrict ourselves to finite-dimensional systems. The Korteweg – de Vries equation can be discretized so as to retain the desired integrability, as was shown by Toda [13] and his collaborators. Another discretization leads to the differential equations duk = 1 (euk+1 − euk−1 ) (k = 1, 2, . . . , n − 1) 2 dt

(1.1)

(where we set formally eu0 = 0, eun = 0) suggested by M. Kac and P. v. Moerbeke [8, 9]. Although this system does not have the appearance of a Hamiltonian system, it can be embedded into one, as was shown in [12]. The remarkable fact is that there are [n/2] = ν polynomials Pµ of uk , euk which are integrals of the motion, i. e., dPµ /dt = 0 (µ = 1, 2, . . . , ν) if one inserts a solution of the above differential equations. Moreover, all solutions can be expressed in the form euk = Rk (η)

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§ 1. Introduction

47

where Rk are rational functions of η = (η1 , . . . , ην )

and η1 = eα1 t , . . . , ην = eαν t .

These rational functions can, of course, not be explicitly described, but this representation suffices to give a complete description of the scattering problem related to this problem (see Section 7). Instead of Calogero’s quantum theoretical problem we look at the corresponding classical one, described by the equations d2 xk = − ∂U , ∂xk dt2 where U=

(k = 1, 2, . . . , n)

 (xk − xl )−2 ,

k, l = 1, 2, . . . , n,

(1.2)

k yn (+∞).

(4.4)

  1 (¨ xn − x (xj − x1 )−3 > 0 (xn − xj )−3 + ¨1 ) = 2

(4.4′ )

t→∞

exists and that From

j1

and the boundedness of x˙ k we conclude, by integration that +∞ (xk − xl )−3 dt < ∞ for k > l = 1 and for l < k = n.

(4.5)

−∞ 1 Note

added in proof. Meanwhile we have been able to verify that indeed δk = 0 for n

q.

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Considering the other differential equations one concludes with a simple induction argument (which we forego) that (4.5) holds for all pairs k > l. This, in turn implies from (3.2) that the limits lim x˙ k (±t) exist, proving (4.3). Because t→∞ of the ordering of the particles we have obviously x˙ 1 (+∞)  x˙ 2 (+∞)  . . .  x˙ n (+∞), x˙ 1 (−∞)  x˙ 2 (−∞)  . . .  x˙ n (−∞).

(4.6)

To prove (4.4) we proceed as follows: Consider first ϕ(t) = xn − x1 > 0 which, by (4.4′), satisfies 1 ϕ¨  2(x − x )−3 > 0. n 1 2

(4.7)

Thus ϕ˙ is monotone increasing and ϕ(+∞) ˙  0, by (4.6). Were ϕ(+∞) ˙ =0 then ϕ(t) ˙ < 0 and thus ϕ bounded. But then the righthand side of (4.7) would be bounded away from zero, hence ϕ unbounded. This contradiction shows that x˙ 1 (+∞) < x˙ n (+∞). Thus, in the first row of (4.6) we do not have equality in all places, i. e., there exists an s with (4.8) x˙ s (+∞) < x˙ s+1 (+∞). From this we will show now x˙ 1 (+∞) < x˙ s (+∞) and x˙ s+1 (+∞) < x˙ n (+∞) which implies readily that all velocity are different. It suffices to show x˙ 1 (+∞) < x˙ s (+∞), the other case being symmetric to it. From (4.8) we conclude that xj − xs = O(t−1 ) for j > s and therefore 1 d2 (x − x ) = (x − x )−3 − O(t−3 ) + (x − x )−3  j 1 s j 1 2 dt2 s j>1

j t0

˙ and is bounded from below. Thus ψ˙ is increasing and ψ(+∞)  0. As before ˙ we conclude that the assumption ψ(∞) = 0 leads to a contradiction. Since ˙ ˙ ψ(t) < ψ(∞) = 0 (t > t0 ) implies ψ to be bounded for t > t0 hence ψ¨ would

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˙ be bounded away from zero, and so ψ unbounded. Thus ψ(∞) > 0 as we wanted to show. Since yk = −x˙ k we have established (4.4). This implies obviously (xk − xl )−1 = O(t−1 )

for t → +∞,

k = l

so that we can see that the matrix L(t) has a limit L(∞) which is a diagonal matrix. Since the eigenvalues λk of L(t) are independent of t we have yk (∞) = λk if we make the convention to order these like λn < λn−1 < . . . < λ1 . for t → −∞ the matrix L also approaches a diagonal matrix with the same eigenvalues in the diagonal, but, because of (4.6) in reversed order. Thus x˙ k (+∞) = −yk (+∞) = −λk ;

x˙ n+1−k (−∞) = −yn+1−k (−∞) = −λk ,

and (4.1) is proven. Finally, we observe that the integrals Ik = Ik (x, y) (k = 1, 2, . . . , n) are in involution. For xk − xk−1 → ∞ these integrals Ik converge with their derivatives to σk (y), the symmetric functions of y. Thus the Poisson bracket Gkl =

n  ∂(Ik , Il ) = {Ik , Il } ∂(xr , yr ) r=1

converges to {σk , σl } = 0. Thus, along any solution of our system Gkl → 0 as t → ∞. On the other hand, as is well known, Gkl are integrals themselves, hence Gkl = 0 for all x, y.

§ 5. The Periodic Case — Sutherland’s Equation If one wants to study the problems of the previous two sections on the circle it is natural to use the identity +∞ 

k=−∞

(x − kπ)−2 = sin−2 x

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as motivation to introduce the potential  U (x) = 1 α2 sin−2 (α(xk − xl )) (α > 0) 2

(5.1)

k=l(n)

where the summation is taken over all distinct pairs k, l (mod n). The coordinates xk of the particles may be defined for all integers k such that xk = xl (mod (π/α))

if and only if k = l (mod n),

so that is suffices to consider xk for k = 1, 2, . . . , n. The differential equations take the form  d2 xk cot α(xk − xj ) sin−2 (α(xk − xj )) = − ∂U = 2α3 2 ∂xk dt j =k(n)

(5.2)

which is the classical analog of Sutherland’s equation [14]. With yk = −x˙ k the Hamiltonian is  2  yk2 + α sin−2 (α(xk − xl )) =1 2 2 k (mod n)

k=l(n)

showing that, on an energy surface = const the minimal distance of the particles remains bounded away from zero and the velocities |yk | bounded away from ∞. Thus the energy surface is compact and most solutions of (5.2) turn out to be quasi-periodic. This will be a consequence of well known facts [1] about integrable Hamiltonian systems if we show that (5.2) has n independent integrals which are in involution. The construction of these integrals follows the pattern of Section 3. We set zkl = α cot α(xk − xl ),

zkl = 0,

if k = l(n),

if k = l(n)

and rewrite the system (5.2) in the form  2 zkj (α2 + zkj ) y˙ k = Uxk = −2 j =k(n)

2 )(yk − yl ) for k = l(n). z˙kl = (α2 + zkl

Here the last line follows from the differential equation for cot x.

(5.3)

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To put these differential equations in the form (3.5) we introduce the n by n matrices 2 Z1 = (zkl ); Z2 = (zkl + α2 ) where k, l = 1, 2, . . . , n. With D2 = diag

 

 2 (zkj + α2 ) ;

j (mod n)

D3 = diag

 

j (mod n)

 2 zkj (zkj + α2 ) ;

Y = diag{yk }. we set L = Y + iZ;

B = iD2 − iZ2 .

(5.4)

Then it is a straightforward, though surprising, calculation that (5.3) can be written in the form dL = BL − LB. (5.5) dt In fact for α → 0 the formal identities go over into those of Section 3, except for the boundary conditions. Thus it follows that the coefficients I1 , I2 , . . . , In det(λI − L) = λn + I1 λn−1 + · · · + In are independent integrals of the motion. We will not verify here that they are involution1, but observe that they are rational functions of yk and eiα(xk −xl ) . To verify (5.5) one has to use the addition theorem for cot x which gives, for k, l, r distinct modulo n: zkl =

zkr zrl − α2 zkr + zrl

hence, for k = l (mod n) 2

Pkl, r = zkr zrl − α − (zkr + zrl )zkl = 1 This



0 −

2 zkl

2

−α

if r = k, l(n)

if r = k, l(n).

could be done by replacing α by iα and using the same argument as in the previous section.

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This implies for 2 2 2 2 Qkl, r = (zkr − zrl )Pkl, r = −zkr (zrl + α2 ) + zrl (zkr + α2 ) − (zkr − zrl )zkl

that 

Qkl, r =

r

⎧ ⎨0

⎩ −2



2 (zkr + α2 )zkr

r

so that the matrix with the elements

 r

if k = l (mod n) if k = l (mod n)

Qkl, r agrees with the diagonal ma-

trix −2D3 . Now we compute the commutator   Qkl, r = −2D3 [Z2 − D2 , Z1 ] = r

and, thus, from (5.4) [B, L] = i[Y, Z2 ] − [D2 , Z1 ] + [Z2 , Z1 ] = i[Y, Z2 ] − 2D3 . From this identity one reads off that (5.5) agrees with the equation (5.3). This makes the statement about the Ik being integrals of the motion again obvious.

§ 6. Rational Character of the Solution of (2.4) We return to the equations (2.4) or (2.5) and investigate their solutions using the fact that these differential equations describe isospectral deformation of Jacobi matrices. We begin with introducing a set of variables rk on the manifold of Jacobi matrices (2.1) for which the spectrum is fixed. This is the analog of the inverse spectrum problem. Let R(λ) = (λI − L)−1 and e1 be the vector with components (1, 0, . . . , 0). We introduce the rational function f (λ) = (R(λ)e1 , e1 ) which has simple poles at λ = λk with a positive residue which we denote by rk2 , so that n  rk2 . f (λ) = λ − λk k=1

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On account of the symmetry property K −1 LK = −L derived in Section 2, f (λ) is an odd function of λ. Thus, if we order the (always distinct) eigenvalues by λn > λn−1 > . . . > λ1

(6.1)

we conclude that λk = −λn−k+1 ;

rk = rn−k+1

and f (λ) can be represented by f (λ) =

ν  2λrk2

k=1

λ2 − λ2k

+ κn

2 rν+1 λ

where κn = 1 for n odd, κn = 0 if n is even and ν = [n/2]. Since f (λ) ∼ λ−1 for |λ| → ∞ we have n 

rk2 = 1.

k=1

and we prefer to free ourselves from the latter restriction by using the rk as projective coordinates. Therefore we set

f (λ) =

n 

k=1

rk2 /(λ − λk ) n 

k=1

rk2

=

ν 

k=1

2 2λrk2 /(λ2 − λ2k ) + κn (rν+1 /λ) ν 

k=1

.

(6.2)

2 2rk2 + κn rν+1

The n variables r1 , r2 , . . . , rν , κn rν+1 , λ1 , λ2 , . . . , λν can be used to describe the Jacobi matrix (2.1) uniquely up to scaling of the rk . In fact, the squares a2k (k = 1, 2, . . . , n − 1) of the elements in (2.1) can be expressed rationally in terms of those rj , λj , 1  j  ν and rν+1 if n is odd. The reason for this fact lies in the representation of f (λ) as a continued function 1

f (λ) = λ−

(6.3)

a21 λ−

a22

..

. λ − a2n−1 λ

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which goes back to Stieltjes (used also in [12]). Since the computation of the continued fraction from the partial fraction expression is a rational process one finds that a2k = Rk (r, λ) (6.4) where the Rk are rational functions, homogeneous of degree zero in the rj and homogeneous of degree two in λ. Moreover, (6.4) can be viewed as mapping which takes the domain D = {(λ, r), λ1 > λ2 > . . . > λν  0; rj > 0 (j = 1, 2, . . . , n − ν)} into the domain onto  = {aj > 0, j = 1, 2, . . . , n − 1} D

in such a way that the pre-image of each point is precisely one ray (ρr, λ) with a scalar ρ > 0. We will show that in these homogeneous coordinates the differential equations take the simple form λ˙ k = 0;

r˙k = −λ2k rk ,

(6.5)

so that, via (6.4) the a2k appear as rational functions of exponentials 2 2 e−λ1 t , . . . , e−λν t . To prove this assertion we introduce the eigenvectors ϕ(λj ) of L which we normalize by (6.6) ϕ1 (λj ) = (e1 , ϕ(λj )) > 0; |ϕ(λj )| = 1. If L is a solution of (2.3) these eigenvectors become functions of which evolve according to ϕ(λj , t) = U (t)ϕ(λj , 0), where U (t) is the unitary matrix of Section 2. Thus the eigenvectors satisfy the differential equation dϕ(λj , t) = −Bϕ(λj , t). dt We compute the resulting differential equation for the first component ϕ1 = = (e1 , ϕ) ϕ˙ 1 = −a1 a2 ϕ3

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63

and use the equations resulting from (L − λ)ϕ = 0

λϕ1 + a1 ϕ2 = 0 a1 ϕ1 − λϕ2 + a2 ϕ3 = 0

to express ϕ3 in terms of ϕ1 . One finds readily

a1 a2 ϕ3 = (λ2 − a21 )ϕ1

so that the differential equations for ϕ1 become

ϕ˙ 1 = −(λ2 − a21 )ϕ1 .

(6.7)

Finally, to show that the ϕ1 (λk ) are proportional to the rk we write the resolvent R(λ) in terms of the eigenvectors obtaining  (ϕ(λk ), e1 )2 f (λ) = (R(λ)e1 , e1 ) = λ − λk k

so that

ϕ1 = (λk ) 

rk n 

j=1

rj2

1/2 .

Thus the differential equations (6.5) give    λ2j rj2 ϕ1 (λk ). ϕ˙ 1 (λk ) = − λ2k − It is easy to verify that

a21

=

 j

λ2j rj2



rj2

j

−1

and the second equations of (6.5) have been verified. The first equations of (6.5) are clear from the derivation. Thus the solutions a2k of (2.4) are rational functions of exponential functions. We describe the solution for n = 4. Computing the continued fraction of f (λ) explicitly one finds (λ22 − λ21 )2 r12 r22 λ2 r2 + λ22 r22 2 , , a = a21 = 1 12 2 r1 + r22 (λ21 r12 + λ22 r22 )(r12 + r22 ) a23 = 2

λ21 λ22 (r12 + r22 ) λ21 r12 + λ22 r22

.

Inserting rj = rj (0)e−λj t we obtain the explicit solutions of (2.4).

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§ 7. The Scattering Problem Associated with the Equation of Kac and Van Moerbeke In order to study the asymptotic behavior of the solution of (2.5) we consider uk = xk − xk+1 ,

k = 1, 2, . . . , n − 1

(7.1)

as the difference between the positions xk of n particles on the line. If the xk satisfy the differential equations x˙ k = − 1 (euk + euk−1 ), 2

k = 1, 2, . . . , n

(7.2)

where we formally set eu0 = 0 = eun , or x0 = −∞, xn+1 = +∞ then clearly (2.5) follows. Conversely the xk are determined only up to translation and for any solution xk (t) of (7.2) also xk (t) + c is a solution giving rise to the same solution of (2.5), provided c is a constant. For simplicity we will assume that n = 2ν is even. We ask for the asymptotic behavior of the solution of (7.2) for t → ±∞ and the relation between the scattering data. We will show that any solution of (7.2) behaves linearly for large t: ± xk (±t) ∼ ±α± k t + βk

as t → ∞

where + − − α+ 2j = α2j−1 = αn−2j+2 = αn−2j+1 ,

j = 1, 2, . . . , ν,

(7.3)

i. e., the particles travel asymptotically in pairs, while the different pairs have negative and different velocities, in fact, it turns out 2 α+ 2j = −2λj ,

j = 1, 2, . . . , ν,

(7.3′ )

where the λ1 > λ2 > . . . are the eigenvalues of L. We will also determine the relation between the phases. First of all, for the neighbors we have the asymptotic distances + + 2 β2j−1 − β2j = log(−2α+ j ) = log(4λj )

− − − βn−2j+2 = log(−2α+ βn−2j+1 j )

and for the phases of pairs with the same velocities   + − + 2 + 2 β2j − βn−2j+2 =− log 4(α+ log 4(α+ 2k − α2j ) + 2k − α2j ) . kj

(7.4)

(7.5)

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7. The Scattering Problem

Thus the particles undergo a scattering in which the pairs behave as if they interacted pairwise at a time. The results (7.3), (7.3′ ), (7.4) are easily derived and we begin with their proof. We recall the differential equation (2.4) a˙ k = ak (a2k+1 − a2k−1 ),

k = 1, 2, . . . , n − 1

with a0 = 0 = an from which we see that n−1 

a2k = const

k=1

along solutions. Thus ak are bounded. Since d log(a a · · · a 2 1 3 2j−1 ) = a2j dt we conclude that

∞ 0

a22j dt < ∞.

Since a˙ 2j is bounded this implies that a2j (t) → 0

as t → +∞.

(7.6)

Thus, the Jacobi matrix L(t), given by (2.1), is asymptotic to a matrix blocked into two by two matrices with eigenvalues ±a2j−1 (t), j = 1, 2, . . . , ν. Since, on the other hand the eigenvalues λk are distinct and independent of t it follows that the limits a2j−1 (t) → a2j−1 (∞) exist and agree with these eigenvalues in some order. From the differential equations a˙ 2j 2 2 a2j = a2j+1 − a2j−1 and from (7.6) it follows that a22j+1 (∞) < a22j−1 (∞) and thus, if we order the eigenvalues λk of L according to (6.1) we conclude a2j−1 (t) → λj ,

(j = 1, 2, . . . , ν).

(7.7)

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Using the relation 4a2k = euk = exk −xk+1

(7.8)

we conclude from (7.1), (7.2), (7.7), (7.8) that x˙ 2j (+∞) = x˙ 2j−1 (+∞) = −2λ2j proving (7.3′ ) and the first part of (7.3). The other part follows by considering the asymptotic behavior for t → −∞ analogously. Moreover, (7.7) and (7.8) implies that x2j−1 − x2j → log(4λ2j )

for t → +∞

proving the first part of (7.4). The second follows similarly. It remains to prove (7.5). This will be done by relating the first order differential equations (7.2) to a second order system related to the Toda lattice, for which the scattering problem has been solved [12]. We notice that differentiation of (7.2) yields x˙ k = − 1 (euk u˙ k + euk−1 u˙ k−1 ) = 2 1 = − {euk (euk+1 − euk−1 ) + euk−1 (euk − euk−2 )} = 4 1 = − (exk −xk+2 − exk−2 −xk ) 4 where we set the undefined exponential terms equal to zero. Thus with ξj = x2j ;

τ = t/2

(7.9)

we have d2 ξj dτ 2

= eξj−1 −ξj − eξj −ξj+1 = ∂U , ∂ξj

where U=

ν−1 

(j = 1, 2, . . . , ν)

(7.10)

eξj −ξj+1 .

j=1

This Hamiltonian system has already been established as an integrable one [13]. For the scattering one has again ′ ξj′ (+∞) = ξν+1−j (−∞),

j = 1, 2, . . . , ν

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References

which is consistent with (7.3) as ξj′ (±∞) = 2α± 2j , and ξj (τ ) − ξν+1−j (−τ ) − 2γj τ →



δkj

(7.11)

k=j

where γj = ξ ′ (+∞) = 2α+ 2j ;

δkj =



log(γk − γj )2, k > j

− log(γk − γj )2, k < j.

With (7.9) the relation (7.11) translates readily into the statement (7.5). We conclude with a comment on the relation between the differential equation (2.5) by Kac and v. Moerbeke and the equations (7.10) for the Toda lattice. The first one corresponds to an isospectral deformation the Jacobi matrix L given by (2.1), with zeros in the diagonal, while the second-order differential equation corresponds to such deformations of such Jacobi matrices with arbitrary diagonal elements (see [4, 12]). To establish the connection between the two we form L2 which is not any more a tridiagonal matrix, but is similar to one. In fact, with eα (α = 1, 2, . . . , n) denoting the unit vectors, one finds that L2 leaves the spaces E1 = span{e1 , e3 , . . . , en−1 } and E2 = span{e2 , e4 , . . . , en } invariant and reduces in each of these spaces to a symmetric Jacobi matrix. This explains why 2 the solutions of (2.4) are rationally expressible in terms of e−λj t while solutions of the corresponding equations for the Toda lattice are rational in e−λj t . This illustrates in a simple example how the operation L → L2 and more generally L → f (L) plays a role in these problems.

References [1] V. I. Arnold, Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimension infinie et ses applications a` l’hydrodynamique, Ann. Inst. Fourier, Grenoble, 1966, 16, 319–361. [2] F. Calogero, Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 1971, 12, 419–436. [3] F. Calogero, C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse square potential, J. Math. Phys., 1974, 15, 1425–1430.

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[4] H. Flaschka, The Toda lattice, I, Phys. Rev., 1974, B9, 1924–1925. [5] H. Flaschka, The Toda lattice, II, Progr. Theor. Phys., 1974, 51, 703–716. [6] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Korteweg – de Vries equations and generalizations, VI, Methods for exact solutions, Comm. Pure Appl. Math., 1974, 27, 97–133. [7] M. H´enon, Integrals of the Toda lattice, Phys. Rev., 1974, B9, 1921–1923. [8] M. Kac, P. van Moerbeke, On an explicitly soluble system of non-linear differential equations related to certain Toda lattices, to appear. [9] M. Kac, P. van Moerbeke, Some probabilistic aspects of scattering theory, to appear. [10] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 1968, 21, 467–490. [11] C. Marchioro, Solution of a three-body scattering problem in one dimension, J. Math. Phys., 11, 1970, 2193–2196. [12] J. Moser, Finitely many mass points on the line under the influence of an exponential potential — An integrable system. (See the article in this collection.) [13] M. Toda, Waves in nonlinear lattice, Progr. Theor. Phys. Suppl., 1970, 45, 174–200. [14] B. Sutherland, Exact results for a quantum many-body problem in one dimension, II, Phys. Rev., 1972, A5, 1372–1376. [15] V. E. Zakharov, and L. D. Faddeev, Korteweg – de Vries equations: A completely integrable Hamiltonian system, Funk. Anal. i Pril., 1971, 5, 18–27.

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Various Aspects of Integrable Hamiltonian Systems1 § 1. Integrable Hamiltonian Systems a) In these informal lecture notes we discuss a number of integrable Hamiltonian systems which have surfaced recently in very different connections. It is our goal to discuss various aspects underlying the integrability of a system like that of group representation, isospectral deformation and geometrical considerations. Since this subject is still far from being understood or being systematic we discuss a number of examples which are seemingly disconnected. In fact, there are some rather unexpected connections like between the inverse square potential of Calogero (Section 4) and the Korteweg de Vries equation. Here we show a surprising new connection between the geodesics on an ellipsoid and Hill’s equation with finite gap potential. b) form

The differential equations of mechanics can be written in Hamiltonian y˙ k = − ∂H ∂xk

x˙ k = ∂H , ∂yk

(k = 1, 2, . . . , n)

(1)

where x = (x1 , . . . , xn ) ∈ Rn , y = (y1 , . . . , yn ) ∈ Rn , are coordinates in the phase space R2n or an open subset of R2n . Thus a function H defines a vector field XH defined by n    ∂H ∂ − ∂H ∂ . XH = ∂yk ∂xk ∂xk ∂yk k=1

For any function F the expression n    ∂H ∂F − ∂H ∂F = {F, H} XH F = ∂yk ∂xk ∂xk ∂yk k=1

is antisymmetric in F , H. It is called the Poisson bracket of F and H. The Hamiltonian systems form a Lie algebra and [XH , XG ] = −X{H, G} . 1 Various

aspects of integrable Hamiltonian systems. Proc. CIME Conf., Bressanone, 1978.

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A nonconstant function F is called an integral of XH if XH F = {F, H} = 0. In particular, H is an integral. If F is an integral of XH then H is an integral of XF . A set of functions F1 , F2 , . . . , Fr are said to be «in involution» or to commute, if {Fk , Fj } = 0 for k, j = 1, 2, . . . , r. This implies clearly that the vector fields XFk commute. If ϕ = ϕ(ξ1 , ξ2 , . . . , ξr ), ψ = ψ(ξ1 , ξ2 , . . . , ξr ) then {ϕ(F1 , . . . , Fr ), ψ(F1 , . . . , Fr )} =

 ∂ϕ ∂ψ {Fk , Fj }. ∂ξk ∂ξj k, j

Thus, if F1 , . . . , Fr are in involution, so are any functions of F1 , . . . , Fr . Definition 1. A Hamiltonian system (1), defined in an open domain D ⊂ R2n is called «integrable» if there exist n integrals F1 , F2 , . . . , Fn in involution with linearly independent gradients, i. e. in D we have (i) {H, Fj } = 0; (ii) {Fk , Fj } = 0, (iii) dF1 , . . . , dFn linearly independent. n  αk (x2k + yk2 ) defines an integrable system in R2n EXAMPLE 1. H = 1 2

k=1

with Fk = x2k + yk2 (k = 1, 2, . . . , n).

EXAMPLE 2. If H = H(y) is independent of x then the system is integrable with Fk = yk . Locally, that is near any point where dH = 0 any system is integrable; in fact, in appropriate canonical coordinates H agrees with y1 which is a special case of Example 2. Generally it makes sense to speak of a system being integrable in a domain which is invariant under the flow generated by XH . It is highly exceptional for a Hamiltonian system to be integrable globally in an invariant open domain — or even locally near a stationary point (where dH = = 0). However many systems occurring in application are closely approximated by integrable systems. For example, the n-body problem becomes integrable in the limit when all but one mass tends to zero. The resulting system is a decoupled system of Kepler problems.

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71

Another example is a system near a stationary point, say x = y = 0 where dH(0) = 0. Assume that the Taylor expansion of H begins with n

 H= 1 αk (x2k + yk2 ) + · · · 2 k=1

with real numbers α1 , α2 , . . . , αn . By a theorem of G. D. Birkhoff one can approximate this system locally by an integrable one (see [3], [4]): Given any large integer N , assume that j1 α1 + · · · + jn αn = 0, |j1 | + |j2 | + · · · + |jn |  N , jk integers implies that j1 = j2 = . . . = jn = 0; then there exist canonical variables x∗1 , . . . , x∗n , y1∗ , . . . , yn∗ such that H = ϕ(F1 , . . . , Fn ) + RN +1 where RN +1 vanishes with derivatives of order  N at the origin and Fj = = x∗j 2 + yj∗ 2 . Thus if we drop the term RN +1 the system is integrable. (To be precise, one has to excise the hyperplanes x∗k = yk∗ = 0 where the dFj are linearly dependent.) On the other hand one can show that in general, even if the α1 , α2 , . . . , αn are rationally independent the system is not integrable in any neighborhood of the origin. c) The structure of the integrable system is particularly simple. Given the integrals one considers the manifolds Nn defined by F1 = c1 , . . . , Fn = cn with appropriate constants c1 , . . . , cn . These manifolds are invariant not only under XH (because of (i) of definition 1) but also under XFj (because of (ii)). Thus XF1 , XF2 , . . . , XFn span the tangent space of Nn . Since these vector fields commute each component of Nn is topologically a cylinder — and in case it is compact, a torus. Thus in the latter case D is foliated by n-dimensional tori. By a theorem due to Arnold [1], [2] and Jost one can near a compact component of Nn introduce canonical coordinates, called x, y again, such that H = H(y1 , . . . , yn ), that y = 0 corresponds to Nn and that points (x, y), ( x, y) with the integer (xj − x j )(2π)−1 correspond to the same points in D. The yk , xk are called the «action-angle» variables, respectively. In other words, the example 2 is typical. In Example 1 these tori are given by x2k + yk2 = ck if the ck are positive. The flows generated by the commuting XF1 , . . . , XFn are the n rotations in the xk , yk plane.

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In action-angle variables the differential equations become x˙ k = Hyk (y),

y˙ k = 0.

Thus the differential equations are linear on Nn . If the frequencies Hy1 , . . . , Hyn are rationally independent then the orbits are dense on Nn . If one solution on Nn is periodic then all are. This occurs if and only if Hyk /jk = ρ is independent of k with some integers j1 , j2 , . . . , jn . Thus for an integrable system the periodic solutions form (n − 1) dimensional families. The proof that Hamiltonian systems generally are not integrable is based on the fact that generically the periodic solutions on a fixed energy surface are isolated.

References [1] V. I. Arnold and A. Avez, Probl`emes Ergodiquies de la M´ecanique classique, Gauthier-Villars, Paris, 1967. [2] V. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974 (Russian). English translation to appear. [3] C. L. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, 1971. [4] J. Moser, Stable and random motions in dynamical systems, Ann. Math. Studies, 77, 1973.

§ 2. Examples of Integrable Systems, Isospectral Deformations In spite of their exceptional character a number of integrable Hamiltonian systems have been discovered recently for which the underlying symmetries are highly unexpected. The most interesting ones are given by partial differential equations and hence of infinite degrees of freedom. For lack of space they will not be discussed here, but some of the following systems can be viewed as discretized versions of those of infinite degrees of freedom. a) Toda lattice. Consider n mass points on the line with coordinates x1 , x2 , . . . , xn and satisfying the differential equations x ¨j = − ∂U ∂xj

(1)

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where U=

n 

exk −xk+1 .

73

(2)

k=1

This system possesses n integrals in involution F1 , . . . , Fn which are rational in yk = x˙ k and exp(xk − xk+1 ). This fact was discovered by Henon and then by Flaschka. The solutions of this system can be expressed in rational form: exk are rational functions of eλ1 t , . . . , eλn t with some distinct constants λk . The original Toda lattice refers to infinitely many particles, but we will restrict ourselves to n < ∞ (see [1]–[4]). b) The inverse square potential of Calogero is also given by (1) but (2) is replaced by  U= (xk − xl )−2 . k 0. Then it follows from x2 y3 − x3 y2 = 0 x3 y1 − x1 y3 = 0

that x3 = y3 = 0 and the problem is reduced to one in R2 × R2 , where we have the quadric x1 y2 −x2 y1 = λ. This problem is still rotation invariant under SO(2) and we may use polar coordinates r, ϕ and conjugate variables pr , pϕ . They can be defined by the canonical transformation x1 = r cos ϕ = Wy1 x2 = r sin ϕ = Wy2

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with W = r(y1 cos ϕ + y2 sin ϕ). Then pr = Wr , pϕ = Wϕ gives pϕ y1 = pr cos ϕ − r sin ϕ pϕ y2 = pr sin ϕ + r cos ϕ and x1 y2 − x2 y1 = λ. Thus pϕ is the integral and H independent of ϕ. The reduced Hamiltonian is  2  = 1 p2r + λ + V (r) H 2 r2

and the ϕ-dependence is obtained separately from ϕ˙ = ∂H = λ2 . ∂pϕ r

This type of reduction was known to Jacobi who «reduced» the 3 body problem in R3 by using the invariance of this system under the Galilei group which contains the rotation group SO(3) as a subgroup. Eliminating the integrals of the center of mass and the angular momentum one can reduce this system of 9 degrees of freedom to one of 4 degrees of freedom, i. e. reduce the phase-space from 18 dimensions to 8 dimensions. Using, in addition, the conservation of energy one has a vector field on a seven dimensional manifold. b) The moment map. We describe the generalization of this reduction in abstract form. We consider a manifold M with a one-form θ for which ω = dθ is nondegenerate, so that (M, ω) is a symplectic manifold. One sometimes refers to (M, θ) as an «exact symplectic» manifold. An example is the cotangent bundle M = T ∗ N of a manifold N with the natural 1-form. If G is a Lie group we speak of a symplectic group action (ϕg ) if for every g ∈ G there exists a diffeomorphism ϕg : M → M such that ϕg ◦ ϕh = ϕgh for g, h ∈ G, ϕe = Id, and ϕg is symplectic, i. e. ϕ∗g ω = ω. If, moreover, ϕ∗g θ = θ we speak of an exact symplectic group action. For G = R this concept is the same as that of symplectic flow. Every such flow is generated by a Hamiltonian vector field, say X for which X ω is closed. We will assume that this one form is even exact and set X ω = dF,

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F being a Hamiltonian for this vector field. For example, in the exact symplectic case we can define F by X θ = F. If A is the Lie algebra of G, and a ∈ A, exp(ta) = g(t) ∈ G then for any function on M — the relation  d f (ϕ ) = Xf g  dt t=0

defines a symplectic vector field on M . This vector field depends linearly on A. If ϕg , hence X are exact symplectic then we can define the corresponding Hamiltonian F = F (p, a) by F = X θ. This Hamiltonian depends linearly on a ∈ A and therefore defines an element ψ ∈ A∗ in the dual of the Lie algebra via F (p, a) = ψ(p), a. The mapping ψ : M → A∗ so defined is called the moment map (of Souriau). These concepts are easily illustrated and motivated by Example 2, where G = SO(3), A = R3 with [a, b] = a ∧ b. The group action on the exact   3  symplectic space (M, θ) = R6 , yi dxi is given by i=1

ϕR (x, y) = (Rx, Ry),

R ∈ SO(3)

and the corresponding vector fields for R = etA , Ax = a ∧ x is obtained by differentiation as X = a1 (x2 ∂x3 − x3 ∂x2 ) + cyclic permutation. This vector field is Hamiltonian with Hamiltonian F =

3  j=1

aj Fj ,

F1 = x2 y3 − x3 y2

(and cyclic perm).

The moment map ψ: R6 → R3 takes p = (x, y) into the angular moment vector (F1 , F2 , F3 ).

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If H0 is invariant under the group action ϕg , i. e. H0 ◦ ϕg = H0 then the corresponding flow ϕt0 generated by H0 leaves ψ invariant, i. e. ψ ◦ ϕt0 = ψ. This generalizes the fact that the angular momentum vector is an integral. The proof of the above statement is immediate (see [3]). d) The coadjoint representation of a group. An important group action, which is independent of symplectic structure, is the adjoint representation of a Lie group G which is given by ϕg : x → g × g −1 = Lg Rg−1 x. Here Lg , Rg denote left and right multiplication. This mapping takes the unit element ∈ G into itself and the linearized map at = e is defined as   Ad (g) = dLg dRg−1  . x=e

It maps A → A and satisfied Ad (g1 g2 ) = Ad (g1 )Ad (g2 ). It is the adjoint representation of G. The induced mapping Ad ∗ (g −1 ) on A∗ , the dual of A, is called the coadjoint representation. For a fixed µ ∈ A∗ one defines the orbits O(µ) of the coadjoint representation as O(µ) = {(Ad ∗ g)µ | g ∈ G}. It is a basic result due to Kirillov and Kostant that these orbits O(µ) carry a symplectic structure. We illustrate this fact with one example, referring for the full discussion to [5]. Let G = GL(n, R), then one sees at once that Ad (T ) : A → T AT −1

where T ∈ G; A ∈ A.

We can identify A∗ with A by representing any linear functional on A by tr(AB) with B ∈ A. Thus we can identify the adjoint and coadjoint representation. In this case the orbits O(µ) are the matrices similar to µ. If µ has distinct eigenvalues the orbit consists precisely of the isospectral matrices. The symplectic structure is defined by giving the values of the two-form on the tangent space of ψ −1 (A) = {T AT −1, T ∈ G}.

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The tangent space at A is given by TA ψ −1 (A) = {[B, A] | B ∈ A}. The differential form is given by ω([B1 , A], [B2 , A]) = tr(A[B1 , B2 ]). One verifies that this form is nondegenerate, skew symmetric and closed, thus defining a symplectic structure on O(A). Consider any two group actions ϕg : M → M and ϕ g : M → M of G. One calls a map τ : M → M equivariant if τ ϕg = ϕ g τ.

If ϕg is an exact symplectic group action then the moment map ψ : M → A∗ is equivariant with respect to ϕg : M → M and the coadjoint representation of A. (This theorem holds under more general assumptions, that is not only if ϕg is exact symplectic.) Thus the image of {ϕg (p) | g ∈ G} for fixed p ∈ M under the moment map ψ is given by the orbit {Ad ∗ (g)µ | g ∈ G}

where µ = ψ(p).

The isotropy group Gµ of µ is defined as the subgroup Gµ = {g ∈ G | Ad ∗ (g)µ = µ}. of G. e) The reduced phase space. Given an exact symplectic group action ϕg : M → M we construct the moment map ψ : A → A∗ and for a fixed µ ∈ A∗ the set ψ −1 (µ) = {p ∈ M | ψ(p) = µ}. This is the subset of M of fixed values of the integral. In the example 2 it consists of those states for which the angular momentum vector is fixed. On this set the group Gµ acts, which in this example consists of all rotations leaving the angular momentum vector µ fixed. If µ = 0 this is a one-dimensional rotation group SO(2). To eliminate this angle of rotation we consider the quotient set ψ −1 (µ)/Gµ which corresponds to elimination of the «ignorable» angle of rotation.

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Under appropriate assumptions (e. g. that ψ −1 (µ) is a manifold, Gµ is compact, that Gµ acts on ψ −1 (µ) fixed point free) we introduce the «reduced phase-space» M = ψ −1 (µ)/Gµ . Thus M is the base space of the bundle π : ψ −1 (µ) → M . This reduced phase space M is a symplectic manifold again and the symplectic structure is given by a two form ω  which is defined as follows: If j : ψ −1 (µ) → M is the injection map then j ∗ ω denotes the pullback of ω to ψ −1 (µ). This form is invariant under Gµ and therefore defines ω  via π∗ ω  = j ∗ ω.

In other words, for V1 , V2 ∈ T ψ −1 (µ) let Vk = (dπ)Vk , then ω  (V1 , V2 ) = ω(V1 , V2 ).

Moreover, if H is a Hamiltonian invariant under ϕg then the «reduced flow»  defined by is given by a Hamiltonian H  ◦ π = H ◦ j. H

In other words H ◦ j, the restriction of H to ψ −1 (µ) is invariant under Gµ and  on M . therefore defines a function H We summarize: If H = H(ϕg ) is invariant under the group action ϕg then (under appropriate assumptions) the phase space can be reduced to M =  = ψ −1 (µ)/Gµ which is a symplectic manifold and the Hamiltonian reduces to H defined by restriction. We do not go into the precise specification of the assumptions since their validity can be easily established in the examples which we discuss. f) Examples. To illustrate this reduction one may consider the Examples 1 and 2, which we leave to the reader. EXAMPLE 3. Consider the group action of G = R (x, y) → (x + sy, y) on R2n , which preserves θ = y, dx. It has the Hamiltonian ψ = 1 |x|2 . 2

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Taking µ = 1 the manifold ψ −1 (µ) is the unit sphere and the isotropy group 2

is G = R. To describe ψ −1 (µ)/G we choose the point of minimal distance in the line x + sy to lie at s = 0, i. e. we choose x, y = 0. Thus ψ −1 (µ)/G ≃ {(x, y) ∈ R2n , |x| = 1, x, y = 0} which is the cotangent bundle T ∗ S n−1 of the unit sphere. One verifies that the one form y, dx projects into the natural 1-form of T ∗ S n−1 . As an example we consider the Hamiltonian H = 1 (|x|2 |y|2 − x, y2 ) 2

(∗)

which is invariant under the above group action. The corresponding vector field x˙ = |x|2 y − x, yx

y˙ = −|y|2 x + x, yy

restricts on T ∗ S n−1 to x˙ = y,

y˙ = −|y|2 x

or x ¨ = −|x| ˙ 2x which is clearly the geodesic flow on the sphere. Thus the geodesic flow appears as the reduced system to H = 1 (|x|2 |y|2 − x, y2 ). Its orbits are circles on the 2 sphere. EXAMPLE 4. The above Hamiltonian (∗) is invariant under the group action (x, y) → (ax + by, cx + dy),

ad − bc = 1

of SL(2, R). Its Lie algebra is generated by the Hamiltonians |x|2 , |y|2 , x, y, and we ask for the reduced space corresponding to |x|2 = 1,

x, y = 0,

|y|2 = 1,

which defines ψ −1 (µ) in this case. We need the isotropy group Gµ which is easily determined to be given by     a b cos ϕ sin ϕ = . c d − sin ϕ cos ϕ

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Note that ψ −1 (µ) is the unit tangent bundle of S n−1 and the orbits of Gµ are precisely the geodesics in the sphere. Thus the reduced space is the orbit manifold of geodesics on the unit tangent bundle. The reduced flow is constant in this case. In the next section we will describe a more complicated example of a reduced phase space under the coadjoint action of the unitary group extended to the cotangent bundle of the Lie algebra. This will lead for appropriate choice of µ to the inverse square potential of Calogero. Actually similar constructions are applicable to the Toda lattice (the group being given by the upper triangular matrices) and the Korteweg – de Vries equation as was shown recently by M. Adler [6].

References [1] V. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974, (to appear in English translation), in particular Appendix 5. [2] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetries, Reports on Math. Physics 5, 1974, 121–130. [3] J. Marsden, Applications to Global Analysis in Mathematical Physics, Publish or Perish, Inc, 1974, in particular Chap. 6. [4] J. M. Souriau, Structure des syst`emes dynamiques, Dunod, Paris, 1970. [5] A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976. [6] M. Adler, On a trace functional for formal pseudodifferential operators and symplectic structure of the Korteweg – de Vries equation, preprint, Univ. Wise., 1978.

§ 4. The Inverse Square Potential a) In connection with his quantum theoretical work Calogero [4] was led to consider the n-body problem on the line. He was led to the conjecture that the corresponding classical problem was integrable which was subsequently proven ([5]). Recently these results were derived most elegantly from a reduction process as it was discussed above applied to the coadjoint representation of the unitary group [1].

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We formulate the results. Consider n distinct points on the real line with coordinates x1 < x2 < . . . < xn , and define the potential U (x) =

 k 0 all solutions are periodic of period 2πa−1 . c) To prove these results we consider the cotangent bundle of the unitary group U (n). We represent its element in the form of a pair (X, Y ) of Hermitian matrices and with the 1-form θ = tr(Y dX). The symplectic structure is defined by the two form ω = tr(dY ∧ dX). In this symplectic space the Hamiltonian system associated with a function H = = H(X, Y ) is given by X˙ = HY ,

Y˙ = −HX

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where HY is the matrix with the entries (∂H/∂Ykj ) etc. The Poisson bracket is given by {F, G} = tr(FX GY − FY GX ).

Therefore it is clear that any two functions F1 = F1 (Y ), F2 = F2 (Y ) independent of X are in involution. Thus any such Hamiltonian is integrable. For us (6) Gp = 1p tr(Y p ). will be relevant. The system belonging to G2 is even linear and has the solution X(t) = X(0) + tY (0), d)

Y (t) = Y (0).

(7)

All these systems defined by Gp are invariant under the group action ϕU : (X, Y ) → (U −1 XU, U −1 Y U ).

The corresponding infinitesimal mapping is (X, Y ) → i([X, A], [Y, A]) where A is Hermitian, i. e. iA belongs to the Lie algebra of U (n). This vector field X˙ = i[X, A], Y˙ = i[Y, A] has the Hamiltonian ψ!A = i tr(A[X, Y ]) = tr(iA[iX, iY ]).

This defines the moment map as

ψ : (X, Y ) → [Y, X] the right-hand side being interpreted as an element of the dual of the Lie algebra. The elements of [Y, X] represent the relevant integrals; [Y, X] is the analogue of the angular momentum vector. To construct the reduced space ψ −1 (µ)/Gµ we choose — with Kazhdan, Kostant and Sternberg the element µ as the matrix ⎛ ⎞ 0 i i ... i ⎜i 0 i . . . i⎟ ⎜ ⎟ ⎜i i 0 . . . i⎟ ⎟ µ = −⎜ (8) ⎜. . . . .⎟ ⎜ ⎟ ⎝i . . . . 0 i⎠ i ... . i 0

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so that ψ −1 (µ) is given by the pairs of Hermitian matrices (X, Y ) for which [X, Y ] = −µ.

(9)

The complete set of solutions is given by the following. Proposition 1. The most general solution of (9) with µ given by (8) is of the form  X = V −1 diag(x1 , . . . , xn )V, (10) Y = V −1 L(x, y)V, where x1 , . . . , xn are distinct real numbers, y = (y1 , . . . , yn ) ∈ Cn and V a unitary matrix satisfying ⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ V e = λe where e = ⎜ . ⎟ . ⎝ .. ⎠ 1

Moreover, the group of the unitary matrices V agrees with the isotropy group Gµ of µ.

Corollary. The quotient space ψ(µ)/Gµ is parametrized by x = = (x1 , . . . , xn ), y = (y1 , . . . , yn ) and to any (X, Y ) ∈ ψ(µ) is by (10) associated a unique pair of matrices (K(x), L(x, y)) where K(x) = diag(x1 , . . . , xn ). The corresponding differential form is computed as ω  = tr(dL(x, y) ∧ dK(x)) =

n 

k=1

dyk ∧ dxk .

PROOF OF THE PROPOSITION. We consider the more general equation

[X, Y ] = i(vk v l ) − ni |v|2 δkl where v = (v1 , . . . , vn ) is any complex vector = 0. The second term is determined so that the trace of the right-hand side vanishes. For vk = 1 the equation goes over into (9).

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We choose a unitary transformation U which diagonalizes X. Note that under X → U −1 XU , Y → U −1 Y U the right-hand side goes over into a similar form where v → U −1 v. Therefore we can analyze the equation for a diagonal matrix X = diag(x1 , . . . , xn ) = K(x). From   1 |v|2 [K(x), Y ]kk = 0 = i |vk |2 − n we conclude that |vk |2 is independent of k. If we specialize to |v|2 = n we get |vk | = 1, or vk = eiθk . Applying the unitary transformation, U = diag(eiθ1 , . . . , eiθn ) we achieve that v = e and the diagonal form of X is not destroyed. Thus we have reduced our system to the case X = K(x), v = e. From [K(x), Y ]kl = (xk − xl )Ykl = i

for k = l

we conclude that the xk are distinct and Ykl =

i xk − xl

for k = l.

If we set Ykk = yk we have the solution X = K(x),

Y = L(x, y)

for the system (9). To find the most general solution we have to investigate those unitary transformations U which leave the equation invariant. Since U −1 (vk v l )U = (wk w l ) where w = U −1 v and since this matrix determines the vector W only up to a factor λ on |λ| = 1 we see that the set of U commuting with (vk v l ) is given by those unitary matrices for which U v = λv. This proves the remaining assertions of the proposition. e)

Thus to any (X, Y ) ∈ ψ −1 (µ) we can associate a unique pair π(X, Y ) = (K(x), L(x, y)) = (U −1 XU, U −1 Y U );

U ∈ Gµ ,

and to any function H(X, Y ) on ψ −1 (µ) invariant under U (n) we can associate a function h(x, y) = π ∗ H = H(K(x), L(x, y)).

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This projection takes the symplectic structure of ψ −1 (µ)/Gµ into the standard n  form dyk ∧ dxk . k=1

In particular, the functions (6) which are in involution go over into Fp = 1p tr Lp (x, y)

which are also in involution. If we observe that 1 tr L2 (x, y) = 1 |y|2 + U (x) 2 2 is the Hamiltonian of the inverse square potential then it is clear that the Fp are rational integrals in involution of this system, proving one part of our theorem. The other part is equally easily derived. The solutions of the G2 -flow are given by (7) which we restrict now to ψ −1 (µ). By the proposition there exist unitary matrices Vt such that (Vt Kt Vt−1 , Vt Lt Vt−1 ) = (V0 (K0 + tL0 )V0−1 , V0 L0 V0−1 ) where we set Kt = K(x(t)), Lt = L(x(t), y(t)). Hence with Ut = Vt−1 V0 we have (Ut−1 Kt Ut , Ut−1 Lt Ut ) = (K0 + tL0 , L0 ). The equality of the second component shows again that tr Lpt = tr Lp0 are integrals, the equality of the first component shows that the xk (t) are eigenvalues of K0 + tL0 , proving the theorem. f) Our derivation was based on the fact that the functions tr Y p are mapped n  to Fp = tr Lp (x, y). Similarly, tr X p is mapped into tr K p = xpk , but here k=1

it is evident that these functions are in involution. Nevertheless, it is useful to consider the map (X, Y ) → (−Y, X)

which carries one set of functions into the other. This mapping is clearly symplectic, preserving ψ −1 (µ). Projecting this mapping to ψ −1 (µ)/Gµ we see that there exists a U ∈ Gµ such that (U −1 K(x)U, U −1 L(x, y)U ) = (−L(ξ, η), K(ξ))

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or



U −1 L(x, y)U = K(ξ), U −1 K(x)U = −L(ξ, η),

91

(11)

where (x, y) → (ξ, η) is the induced mapping. Notice that this mapping is a symplectic algebraic mapping of {x1 < x2 < . . . < xn , y ∈ Rn } onto {ξ1 < ξ2 < . . . < ξn , η ∈ Rn }. It takes our Hamiltonian 1 tr L2 (x, y) into 2

n

1 tr K 2 (ξ) = 1  ξ 2 k 2 2 k=1

and therefore the differential equations into the linear ones ξ˙ = 0, η˙ = −ξ. g) Scattering map. If we observe that in our problem the particles repel each other one sees that the solution runs apart and has asymptotic behavior, xk (t) = αk t + βk + O(t−1 ) yk (t) = αk + O(t

−1

)

for t → +∞

(12)

where the αk are distinct. If we insert this estimate in L(x(t), y(t)) and recall that its eigenvalues are independent of t then we see that α1 , . . . , αn are the eigenvalues of L(x(t), y(t)), in particular of L(x, y) where x = x(0), y = = y(0). Thus in (11) we can identify the ξj with the asymptotic velocities yj (∞). Similarly, by inserting the flows in the second equation of (11) we get Ut−1 K(x(t))Ut = −L(ξ, η − tξ) = L(−ξ, tξ − η) from which we read off that −ηj = βj . Thus with the mapping (x, y) → (ξ, η) defined by (11) we have xk (t) = ξk t − ηk + O(t−1 ), yk (t) = ξk + O(t−1 )

for t → +∞

where xk , yk are the initial values. Thus (11) is the scattering mapping of the initial values (x, y) into the asymptotic velocities ξ and phases −η. For t → −∞ one obtains similarly asymptotic formulas, say xk = ξk− t − ηk− + O(t−1 ) for t → −∞.

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Then it follows at once from the reversibility of the system that − ηk = ηn−k+1

− ξk = ξn−k+1 ,

showing that the scattering of this system is the same as for elastic reflections. EXERCISE 1. Show that for any 2 function fj the functions tr fj (X 2 + Y 2 ) (j = 1, 2) are in involution. The same holds for tr fj (XY ). EXERCISE 2. Show that under the above reduction the Hamiltonian H = 1 tr(Y 2 + a2 X 2 ) 2 goes over into the Hamiltonian 1 |y|2 + 1 a2 |x|2 + (x − x )−2 . k j 2 2 k 0 into slit domain obtained by deleting vertical slits at Re ω = θ( λ2j−1 ) = θ( λ2j ) (Schwarz – Christoffel’s formula). # # We have to choose the parameters λj , λ′j so that θ( λ2j−1 ) = θ( λ2j ) = = jπ, so that ∆(λ2j ) = 2 cos jπ = 2(−1)j . Thus we can parametrize the spectrum most effectively by choosing N positive numbers h1 , h2 , . . . , hN and

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119

define θ(z) as the unique schlicht conformal mapping of Im z > 0 onto the domain obtained from Im w > 0 by deleting the slits 0 < Im w  hj , j = 1, . . . , N $ # ′ = ± λ2j , z±j = ± λ′j

Re w = ±jπ, and such that, with z±j

θ(z±j ) = ±πj,

′ θ(z±j ) = ±πj + ihj

for j = 1, 2, . . . , N ; and θ(0) = 0. Moreover, for large values of z we have Q(z) ∼ z. This shows that the 2N numbers λ1 , . . . , λ2N cannot be chosen arbitrarily, but depend only on the N parameters h1 , . . . , hN . Given the spectrum, however, also the λ′1 , λ′2 , . . . , λ′N as well as the other eigenvalues λj (j > 2N ) are uniquely determined. Consequently, the spectrum λ0 = 0, λ1 , . . . , λ2N determines λ′j and all the double eigenvalues, hence the discriminant ∆(λ) = 2 cos θ(λ). This takes care of (A). d) Description of the potential in terms of an auxiliary spectrum. We ask for the set of all potentials belonging to a given spectrum of the type (7). To fix a potential one may use another spectrum, e. g. the eigenvalues belonging to the boundary conditions (3) (actually this gives rise to 2N potentials in general). In this part we will be sketchy and refer to the paper (e. g. Trubowitz [6]). We mentioned that for any potential q(x) = q(x + 1) the eigenvalues µj for (3) lie in the interval λ2j−1  µj  λ2j . Conversely we wish to construct a potential q for arbitrary eigenvalues µj in the above interval, where the λj are given according to (7). We can compute q(0) from (6). In order to find q(x) we subject q to the translation q(x) → q(x + t) so that the eigenvalues µj are taken into µj (t) in the above interval. We will derive a differential equation for these functions µj (t) and integrate them; thus we can recover q(t) from the formula (6) in a unique way. Below we will derive these differential equations which show that for increasing t the eigenvalues oscillate in the interval λ2j−1  µj  λ2j back and forth, the double valuedness coming from the unspecified # sign of √ ∆2 − 4. We can make the choice unique if we assign the sign of ∆(µ)2 − 4

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to each point, effectively making each interval into # a circle. For any specification of µj in λ2j−1  µj  λ2j and a sign of ∆(µ)2 − 4 one finds a unique potential this way. A complete proof would require the verification that q(t) so determined is of period 1. We forego this proof and turn to the determination of the differential equation. e) The differential equations for q(x) → q(x + t). If we replace q(x) by q(x + t) we replace the parameters µ1 , . . . , µN by µ1 (t), . . . , µN (t) and we ask for the differential equations describing this flow. We will write this differential equation in two different forms: In explicit form it has the form # N  2 −R(µj ) dµj where A(z) = (µj − z) (8) = dt A′ (µj ) j=1 or in the implicit form N  j=1

µpj

dµj = # 2 −R(µj ) dt



0 for 1 for

p = 0, 1, . . . , N − 2, p = N − 1.

(9)

The latter follows from the former by the observation that the left-hand side is the residue sum for the integral z p dz 1 2πi A(z) taken over a large circle. These are the differential equations asked for under (c). The latter form exhibits the familiar Abelian sums of the differentials µp dµ # 2 −R(µ)

of the first kind on the Riemann surface defined by w2 = −R(z). We derive the above differential equations (see E. Trubowitz): We view the eigenvalues µj as functionals of q and compute the differential δµj = ϕ2j δq where ϕj is the normalized eigenfunction of Lϕj =

−ϕ′′j

+ qϕj = µj ϕj ;

1 0

ϕ2j dx = 1.

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This is easily verified by taking the differential of this relation with respect to q, (L − µj )δϕj + ϕj δq = δµj ϕj . Taking the inner product with ϕj we get 1 0

ϕj (L − µj )δϕj dx +

1

ϕ2j δq dx = δµj .

0

Noting that the first term vanishes as L is selfadjoint, we have the stated relation. Therefore we have for the translation the differential equation dµj = dt

1

δµj dq dx = δq dt

0

1

ϕ2j (x)q ′ (x) dx

0

= −2

1

ϕ′j ϕj q dx.

0

Using the differential equation we replace ϕj q by ϕ′′j + µj ϕj and obtain 1 dµj  = −ϕ′j 2  = (ϕ′j (0))2 − (ϕ′j (1))2 . dt 0

Clearly, ϕj (x) is a multiple of the solution y2 (x, µj ) which also vanishes at x = 0. One computes ϕj (x) = (y˙ 2 (1, µj )y2′ (1, µj ))−1/2 y2 (x, µj ) where the dot indicates λ-derivative. Thus (ϕ′j (0))2 − (ϕ′j (1))2 = y˙ 2 (1, µj )−1 (y2′ (1, µj )−1 − y2′ (1, µj )). If we use the relation y1 y2′ − y1′ y2 = 1 and the fact that for λ = µj , and x = 1 we have y2 = 0 we find y1 y2′ = 1. Therefore we have the identity (y1 + y2′ )2 + (y1 − y2′ )2 = 4y1 y2′ = 4 for x = 1, λ = µj . Hence, with ∆ = y1 + y2′ 4 − ∆2 = (y1 − y2′ )2 = (y2′ −1 − y2′ )

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$ dµj = ϕ2j (0) − ϕ2j (1) = y˙ 2 (1, µj )−1 4 − ∆2 (µj ). dt

Since y2 (1, λ) has as zeroes the µk (k = 1, 2, . . .) and (4 − ∆2 (λ)) as zeroes the λ0 , λ1 , . . . we obtain the differential equation (8) to a constant. We forego the determination of the constant, which is found from the asymptotic behavior for large λ. In fact, Trubowitz uses these differential equations in the infinite dimensional case in the form   µj − µk −1 # dµk = k2 π2 ∆2 (µk ) − 4 2 2 dt j π j =k which is valid in the general case and uses them to solve the inverse spectral problem in the periodic case. The point is that the right-hand side admits a uniform Lipschitz estimate to allow global solution of the differential equation. However, we restrict ourselves to the finite gap case. For example, if N = 1 the differential equation for µ = µ1 becomes # dµ = 2 (λ0 − µ)(λ1 − µ)(λ2 − µ) dt

which gives the elliptic p-function plus a constant. Formula (6′ ) shows that the corresponding potential is also an elliptic function with real period 1. This is the same equation. f) A mechanical description of the differential equations (8) or (9). McKean and Trubowitz derived an interesting identity for the eigenfunctions ϕ2j belonging to the eigenvalues λ2j . They satisfy the relation ∞ 

εj ϕ22j (x) = 1

j=0

where εj =

∆(λ2j ) . λ2j )

y1′ (1,

(10)

The εj depend on the λ0 , λ1 , . . . only and are positive if λ2j − λ2j−1 > 0, and are equal to zero if λ2j − λ2j−1 = 0. Thus in our case (7) of finitely many gaps this identity reduces to N  εj ϕ22j (x) = 1. (10′ ) j=1

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Setting xj =

√ εj ϕ2j

(j = 0, 1, . . . , N )

we see that the x = (x0 , x1 , . . . , xN ) are restricted to the unit sphere. We wish to derive the differential equations on the unit sphere which correspond to the translation q(x) → q(x + t). For this purpose we use the differential equations for ϕ2j which give d2 x = √ε ϕ′′ = (g − λ )√ε ϕ = (q − λ )x . j j 2j 2j j 2j 2j j dt2

(11)

2

The fact that |x| = 1 is invariant under the flow implies that x, x ˙ = 0 and

x, x ¨ + |x| ˙ 2 = 0. Thus, taking the inner production with x we obtain from the differential equation: N  −|x| ˙ 2 = q|x|2 − λ2j x2j j=0

or q=

N  j=0

λ2j x2j − |x| ˙ 2.

(12)

We notice that this system (11), (12) is precisely the integrable system of the particle moving on the sphere under the influence of a quadratic potential discussed in a previous section. Here q plays the role of the normal force and N  λ2j xj of the quadratic potential. We can make use of our information about

j=0

the mechanical problem to gain information about the spectral problem. For example, (12) yields the identity q(x) =

N  j=0

(λ2j εj ϕ22j − εj ϕ′2j 2 )

for the potential. In McKean – Trubowitz (p. 223) one finds another formula for of the form q(x) = 2

N  j=0

λ2j εj ϕ22j + λ0 −

N  j=1

(λ2j − λ2j−1 ).

(13)

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Taking the difference of the two expressions we find N 

εj ϕ′2j 2 +

j=0

N  j=0

λ2j εj ϕ22j = −λ0 +

Now, the left-hand side is |x| ˙2+

N 

j=0

N  j=1

(λ2j − λ2j−1 ).

λ2j x2j , i. e. twice the total energy for

the mechanical problem, which is clearly a constant. Its value is given by the right-hand side. g) Identification of the torus T N . Thus we see that the differential equation of translation q(x) → q(x + t) agree with the mechanical problem x¨j = (q − λ2j )xj q as in (12). But clearly not all solutions give rise to periodic potentials. Which solutions correspond to the N -gap potentials described by (7)? They form an N -dimensional torus which we wish to identify. For this purpose we use the previously discussed integrals Fν = x2ν +

 (xν yµ − xµ yν )2  (ϕ2ν ϕ′2µ − ϕ2µ ϕ′2ν )2 = εν ϕ22ν + εν εµ λ2ν − λ2µ λ2ν − λ2µ

µ=ν

µ=ν

and the corresponding function Φz =

N 

ν=0

Fν . λ2ν − z

Since the integrals are in involution it is clear that the manifolds Fν = cν are tori, provided they are regular, compact and connected. Therefore it is to be expected that the desired manifold corresponding to the N -gap potential is described in the algebraic form Fν = cν (ν = 0, 1, . . . , N ). This is indeed the case. But instead of giving the values of the constants cν we describe the zeroes of Φz instead. Since the asymptotically N  1 εν ϕ22ν + O(|z|−2 ) = −z −1 + O(|z|−2 ) Φz = − z ν=0

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we have N 

j=1

Φz = −

N 

(z − uj )

(z − λ2ν )

ν=0

where uj are the zeroes of Φz . Thus Φz is characterized by the zeroes and poles. Theorem. The finite gap potentials q(x) defined by (7) are characterized by uj = λ2j−1 (j = 1, . . . , N ) i. e.

Φz = −

N 

j=1

(z − λ2j−1 )

N 

.

(z − λ2ν )

ν=0

This follows from a scrutiny of Proposition 1, p. 175 of McKean and Trubowitz. Corollary 1. For a given simple spectrum λ0 , λ1 , . . . , λ2N the function Φz is well defined, i. e. Fk have specified values, defining an N -dimensional torus N  on the tangent bundle of εν ϕ22ν = 1. All solutions on this torus are periodic ν=0

of period 1 defining the potentials q(x + t).

Corollary 2. Since the λ2j−1 interlace the λ2j it follows that on this torus all the Fk have positive values. All solutions ϕ2ν are hyperelliptic functions of x and so are the corresponding potentials according to (13), both periodic of period 2 or 1. One can determine a particular potential with given simple spectrum λ0 , λ1 , . . . , λ2N by setting µj = λ2j−1 . (In McKean – Trubowitz this is chosen as the «origin» on the torus T N .) Then q(x) is an even function and the λ2j−1 = µj are roots of y2 (1, λ) while the λ2j are the roots of y1′ (1, λ). Therefore for this choice of the potential q(x) one has Φz =

y2 (1, z) . y1′ (1, z)

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Since the roots of y2 (1, z), y1′ (1, z) in λ2j−1  λ  λ2j coalesce if j > N the above expression is in fact a rational function. It is suggestive to investigate the potentials belonging to other tori, i. e. to different values of the constants cj = Fj . All these solutions are hyperelliptic, but not in general quasi-periodic. Since very little is known about the spectral theory of quasi-periodic potentials it would be worthwhile investigating even these special examples which present themselves through this surprising connection between Hill’s equation and the mechanical problem. (This connection was found by E. Trubowitz and the author.) However, this approach has not yet been carried out.

References [1] A. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg – de Vries type, finite zone linear operators and Abelian varieties, Russ. Math. Surveys 31 (1976) 59–146. [2] V. A. Marˇcenko, and I. V. Ostrovskii, A characterization of the spectrum of Hill’s operator, Mat. Sbornik 97 (139) 1975, 493–554. [3] H. Hochstadt, On the determination of Hill’s equation from its spectrum, Arch. Rat. Mech. Anal., Vol. 19 (1965) 353–362. [4] H. P. McKean, and P. van Moerbeke, The spectrum of Hill’s equation, Inventiones Math. 30 (1975) 217–274. [5] H. P. McKean, and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976) 14–226. [6] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977) 321–337. [7] N. Levinson, The inverse Sturm – Liouville problem, Mat. Tidsskr. B. (1949) 25–30.

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Geometry of Quadrics and Spectral Theory1 § 1. Introduction a. Background In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary differential equations described by a Hamiltonian function and possessing sufficiently many integrals (or conserved quantities) so that they are more or less explicitly solvable by quadrature. Therefore these systems played a crucial role in the last century before more qualitative methods for differential equations were developed at the turn of the century. Subsequently interest In these systems decreased, partly due to the realization that the existence of global integrals can be established only for exceptional Hamiltonian systems. In the last 15 years the subject of integrable Hamiltonian system has regained considerable interest with the discovery of some partial differential equations which can be viewed as such systems with infinite degrees of freedom. In this case the integrals form an infinite sequence of conserved functionals. The most celebrated example is the Korteweg – de Vries equation: ut + uux + uxxx = = 0. Extensive investigations of this equation have led to surprising links with scattering theory, spectral theory, complex analysis of hyperelliptic curves and their θ-functions, and differential geometry. The purpose of this paper is to establish a connection of some classical integrable Hamiltonian systems with the elementary geometry of quadrics. The motivation starts with the following observation: The classical approach to finding the relevant integrals was based on solving the Hamilton – Jacobi equation by separation of variables (St¨ackel [19], Jacobi [6]). This required the appropriate choice of variables and computational skill. A case in point is Jacobi’s integration of the geodesics on an ellipsoid or C. Neumann’s study [14] of a mass point moving on a sphere under the influence of a linear force. On the other hand, in the recent studies of partial differential equation the integrals were found as eigenvalues of some linear operators which depend 1 Geometry of quadrics and spectral theory. The CHERN Symposium 1979, Berkely, Springer Verlag New York, 1980, 147–187.

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on the solution of the partial differential equation but have the feature that their spectrum is conserved for each solution of the partial differential equation considered. Thus under the time evoluation of this equation the linear operator changes in such a way that its spectrum remains fixed, i. e., it undergoes an isospectral deformation. The eigenvalues, viewed as functionals, represent the integrals. This approach of using isospectral deformation of a linear operator has been developed by P. D. Lax in connection with the Korteweg – de Vries equation and has been applied by other investigators to many other examples. The question arises naturally whether all integrable Hamiltonian systems can be described by isospectral deformation. The question is shifted from finding the integrals of the systems, provided they exist, to finding the linear operator whose spectrum is preserved. We will not attempt to answer this question in any generality, but consider some classical examples, such as Jacobi’s geodesic flow on the ellipsoid, and construct an isospectral deformation for them. The relevant matrix turns out to be symmetric, and we will give a geometrical interpretation for the eigenvalues and eigenvectors. This does not lead to new results for this old problem, but to an interesting geometrical interpretation of the eigenvalues and eigenvectors of these operators. In the course of this investigation we will see that our approach also is applicable to the Korteweg – de Vries equation, thus establishing a link between this partial differential equation and the theory of confocal quadrics. b. Geodesics on an Ellipsoid We begin directly to illustrate our approach with the geodesic flow on an ellipsoid, which had been first integrated by Jacobi. In December 1838 he wrote to his friend and colleague Bessel: «Yesterday I solved the equations for the geodesic lines on an ellipsoid with three different axes by quadrature. These are the simplest formulae of the world, Abelian integrals, which turn into elliptic integrals if two of the axes become equal.» This quotation shows how much the Abelian integrals were m vogue at the time; below we will see how this theory ties in with isospectral manifolds. If A is a positive definite symmetric n by n matrix with distinct eigenvalues and x ∈ Rn an n-vector, then we write the equation for the (n − 1)-dimensional ellipsoid as (1.1)

A−1 x, x = 1 and the differential equations of the geodesics as −1 d2 x = −νA−1 x, ν = A y, y , y = dx , (1.2) dt dt2 |A−1 x|2

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where we restrict ourselves to solutions which lie on the ellipsoid. Here ·, · denotes the inner product in Rn . For this problem it will turn out that the relevant isospectral matrices L, which we give here without motivation, are of the form L(x, y) = Py (A − x ⊗ x)Py ,

(1.3)

where the tenser product x ⊗ y denotes the matrix (xi yj ), and Py the orthogonal projection into the orthogonal complement of the vector y. Thus the symmetric matrix L(x, y) depends on two vectors x, y ∈ Rn , where, however, the length of y = 0 is irrelevant. If we identify x with the position on the ellipsoid and set y = dx/dt, then the eigenvalues λ1 , λ2 , . . . , λn of L are preserved under the geodesic flow (1.2). Actually one eigenvalue, say λn is equal to zero and belongs to the eigenvector y of L. But the other n−1 eigenvalues are nontrivial algebraic integrals of (1.2). It is better to form the symmetric functions of the λj and look at the characteristic polynomial l(z) = det(zI − L) of L, which is a polynomial of x, y. In fact, the ratio |y|2 det(zI − L) (1.4) z det(zI − A) = Φz (x, y) is a rational function of z with poles at the eigenvalues α1 , α2 , . . . , αn of A and zeros at λ1 , . . . , λn−1 the nontrivial eigenvalues of L(x, y). As a function of x, y the function Φz (x, y) is a quartic polynomial. The partial-fraction expansion of Φz (x, y) is Φz =

n  Gj (x, y) , z − αj j=1

where the Gj (x, y) are also quartic polynomials of x, y which are integrals for the flow (1.2). Actually only n − 1 of them are independent on the ellipsoid, since there the relation n  α−1 0 = Φ0 = − j Gj (x, y) j=1

holds. We wish to indicate the connection with confocal quadrics to the ellipsoid (1.1), which are given by the equation

(z − A)−1 x, x + 1 = 0.

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We will set Qz (x, y) = (z − A)−1 x, y,

Qz (x) = Qz (x, x),

(1.5)

and denote the quadric Qz (z) + 1 = 0 by Uz . To interpret geometrically the eigenvalue equation Φz (x, y) = 0 of (1.4) we first establish the identity Φz (x, y) = Qz (y)(1 + Qz (x)) − Q2z (x, y), so that for fixed z, x this represents a quadratic form. The equation Φz (x, y) = 0 represents the quadratic cone of tangents to Uz , going through the point x, after the point x is translated to the origin. Secondly one has Φz (x + sy, y) = Φz (x, y), so that Φz is constant along any line x = x0 + sy, y = 0. From these facts one sees that for a given line x = x0 + sy, the roots z = λ1 , λ2 , . . . , λn−1 of the equation Φz (x0 , y) = 0 are such that the above line is tangent to the confocal quadrics Uλj (j = 1, 2, . . . , n − 1). Generically a line in Rn touches just n − 1 confocal quadrics — and the set of lines tangent to Uλ1 , . . . , Uλn−1 forms a normal congruence; see Bianchi [2]. Thus the «isospectral» manifold of matrices L(x, y) with a fixed distinct spectrum λ1 , λ2 , . . . , λn−1 is identified with the normal congruence of common tangents to n − 1 confocal quadrics Uλj (j = 1, 2, . . . , n − 1), which can be considered as a geometrical interpretation of the spectrum of L(x, y). Also the eigenvectors ϕj of L have a simple geometrical interpretation: The eigenvalue λn = 0 corresponds to ϕn = y as was mentioned above, while the other eigenvectors ϕj are the normals to Uλj at the point of contact of the line x = x0 + sy. Since L = L(x, y) is symmetric, the n vectors are pairwise orthogonal — which is the content of an old theorem of Chasles (Bianchi [2],

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Salmon and Fiedler [17]). Under the geodesic flow the orthonormal frame ϕj will undergo a motion described by an antisymmetric matrix B, so that ϕ˙ j = Bϕj ,

L˙ = [B, L].

This is the Lax representation of the geodesic flow, where B turns out to be the matrix −1 B = −(α−1 i αj (xi yj − xj yi )). The fact that the eigenvalues λj are preserved under the geodesic flow means obviously that the tangents of one geodesic of an ellipsoid will touch the same n − 2 quadrics confocal to the ellipsoid — also a well-known result of geometry. These results will be derived in Section 3. There are further properties of the isospectral manifold M(λ1 , . . . , λn−1 ) of matrices L(x, y) with fixed spectrum, which will be established in Section 4. If we identify the lines x = x0 + sy on M(λ) to points and also take the quotient under the reflections xj → ±xj , then we arrive at an (n − 1)-dimensional manifold M′ (λ) which is isomorphic to the Jacobi variety of the hyperelliptic curve w2 = P2n−1 (z) = z −1 det(zI − L) det(zI − A), which is of genus n−1. Thus the Jacobi variety has complex dimension n−1 and is a torus with 2n − 2 periods. The geodesic flow is linear in the variables of the Jacobi map, and thus the geodesic flow is closely related to Abel’s theorem for hyperelliptic integrals. This fact was used by Staude [20] to give a geometrical interpretation of the addition theorem for hyperelliptic integrals. c. Perturbations of Rank 2 In the above approach the choice of the matrices (1.3) was unmotivated and it is difficult to make the right guess. At present there seems to be no systematic way for finding such isospectral matrices. In this case I owe the essential hint to M. Adler, who suggested looking at matrices of the form A + x ⊗ y − y ⊗ x. The matrices (1.3) can be obtained as limit case of similar matrices, which we will now discuss. If A is again a fixed symmetric matrix and x, y, ξ, η four n-vectors, we call A+x⊗ξ+y⊗η

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a rank-2 perturbation of A. We will study the special case where ξ = ax + by,

η = cx + dy,

so that L(x, y) = A + ax ⊗ x + bx ⊗ y + cy ⊗ x + dy ⊗ y

(1.6)

is a matrix which depends on two n-vectors x, y while a, b, c, d are fixed with ∆ = ad − bc = 0. We will take this 2n-parameter family of matrices as starting point, study the algebraic manifold M(λ1 , λ2 , . . . , λn ) of those x, y ∈ Rn for which L(x, y) has the fixed spectrum λ1 , λ2 , . . . , λn , and investigate the isospectral deformations of these matrices. The basic observation is the following: If we consider the symplectic manifold (R2n , ω) with the symplectic two-form ω=

n  j=1

dyj ∧ dxj ,

then the eigenvalues of L(x, y) given by (1.6) are in involution, {λj , λk } = 0, where {F, G} =



(Fxj Gyj − Fyj Gxj )

denotes the standard Poisson bracket. Again it is better to use the symmetric functions of the eigenvalues of the λj or the functions Φz (x, y) = 1 −

det(zI − L) , det(zI − A)

which are quartic polynomials in x, y. With the partial-fraction expansion n  Gj (x, y) Φz (x, y) = , z − αj j=1

we have n quartic polynomials Gj which are in involution. The Hamiltonian vector fields XH for any Hamiltonian H = = ϕ(G1 , G2 , . . . , Gn ) — or any Hamiltonian depending on the spectrum of L only — is tangential to M(λ), and XH =

n  ∂H · X , Gj ∂Gj j=1

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where because of [XGj , XGk ] = −X{Gj , Gk } = 0 all these vector fields commute. In particular, the isospectral manifolds are Lagrange manifolds. All these Hamiltonian systems are integrable, by which we mean a vector field having n integrals Gj in involution, for which dGj are linearly independent in an open dense set. In Section 2 we will show how these Hamiltonian vector fields can be written in the form d L = [B, L]. dt In Section 3 it will be shown that the geodesic flow on an ellipsoid can be derived as the limit case of matrices (1.6) with a = 0, b = −c = ν, d = ν 2 for ν → ∞. d. Hyperelliptic Curve In Section 4 we show how M(λ) is related to the Jacobi variety of a hyperelliptic curve. The manifold M(λ) is n-dimensional, but after factoring out a 1-dimensional group one is led to an (n − 1)-dimensional manifold M′ which is isomorphic to the Jacobi variety of a hyperelliptic curve of genus n − 1. This generalizes the statement for the geodesic flow on the ellipsoid in which case M′ is obtained by identifying the straight lines x → x + sy to points and factoring out the reflections xj → ±xj . The proof is based on introducing a second matrix M = M (y) whose spectrum µj together with that of L = L(x, y) determines x, y. In other words, x, y are described by two spectra, each of which is given by a set of functions in involution. The spectrum of M is viewed as a divisor in the Jacobi map. The computation of the symplectic form ω in these variables takes the form  ω= Sλj µk dλj ∧ dµk ,

where S = S(λ, µ) is the function to which one is also led by the separation of variables of the Hamilton – Jacobi equations. In this way one sees the connection of the two approaches. For details we refer to Section 41 . e. Applications In Section 5 we describe various classical integrable examples for which the integrals can be obtained in terms of the eigenvalues of matrices of the 1 ln this connection we refer to the forthcoming «Lectures on θ-functions and Their Applications» by David Mumford, held at Bombay and Harvard 1978–1979.

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form (1.6). Here we have to distinguish three cases according as the rank of the symmetric part ⎛ ⎞ b+c a 2 ⎠ ⎝ b+c d 2

a b

of the matrix c d is 2, 1, or 0. We give three examples, the last one being illustrated by a subclass of geodesics of the orthogonal group with a left-invariant metric, as it was studied first by Arnold. (For literature see Dikii [4], Manakov [8], Mischenko [11].) Finally we show how these systems relate to the finite band potential in the Hill’s equation, in the periodic and the quasiperiodic case. This is based on a connection between the translation flow for the above finite-band potentials and a mechanical problem of a particle moving on the sphere |x| = 1 under the influence of a linear force. This connection was found by E. Trubowitz; it was described previously (Moser [12]). Finally, in the Appendix we describe a class of matrices L involving x−1 j whose eigenvalues are also in involution. They are also integrals for a classical mechanical system discussed in the dissertation of Rosochatius [16]. f. Connection with M. Reid’s Result [15] We want to mention a related result which we learned from a letter of Horst Kn¨orrer. In his unpublished dissertation of 1972 Miles Reid established that the set of (m − 1)-dimensional linear subspaces of a nonsingular intersection of two quadrics in P2m+1 (C) is — as algebraic manifold — isomorphic to the Jacobi variety of a hyperelliptic curve. It is tempting to guess a connection to the above result about the common tangents of n − 1 confocal quadrics Uλ1 , . . . , Uλn−1 in Cn . Such a connection really exists, and Kn¨orrer communicated to me a beautiful construction of a 1-to-2n−1mapping of the variety of common tangents to M. Reid’s1 Jacobi variety for some appropriate quadrics. g. Final Remarks The above approach is obviously very unsystematic and relies on lengthy calculation. Why are the eigenvalues of matrices of the form (1.6) in involution? The deeper reasons have still to be revealed. M. Adler, who gave the 1 Note

added in proof : H. Kn¨orrer on the ellipsoid, Inv. Math. 59, 119–143 (1980).

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initial hint for the form of the isospectral matrices required, found a general framework based on the coadjoint representation of certain Kac – Moody algebras in extension of his previous work [1], which allow him to encompass the above examples as special cases of a general theory. Originally it was the plan to publish a joint paper from this point of view; however, since the theory is formidable and lengthy, it was necessary to separate off the general approach. Adler’s Lie-algebraic approach will appear elsewhere. Here we merely want to show the baffling connection between the spectral theory of the matrices (1.6) and the geometry of quadrics. I want to express my thanks to Horst Kn¨orrer for informing me about his geometrical construction in connection with M. Reid’s work. After presenting this paper in Berkeley, I visited in Warwick and lectured on this topic. I want to thank D. Epstein for his hospitality and Adrian Douaday for interesting discussions. He supplied an elegant alternative proof for the involuntary character of the eigenvalues of (1.6). Because of length restrictions we could not present his argument here. I am grateful to P. Deift for suggestions and for reading the manuscript. Finally, I am particularly indebted to M. Adler, who contributed essential ideas to this work.

§ 2. Perturbation of Rank 2 a. Isospectral Manifolds We take as a starting point the spectral problem for a perturbation of rank 2 of a symmetric bilinear operator. Let V denote a real (or complex) finite-dimensional vector space, ,  a real inner product, and let A be a matrix symmetric with respect to the inner product, i. e. Av, w = v, Aw. Moreover, we will assume the eigenvalues of A to be distinct. A perturbation of rank r is given by Lv = Av +

r  ρ=1

xρ ξρ , v,

where x1 , . . . , xr and ξ1 , . . . , ξr are two sets of linearly independent vectors in V . We write the above formula in terms of the tensor product ⊗ as L= A+

r  ρ=1

xρ ⊗ ξρ .

(2.1)

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It is well known that the spectrum of L is determined by the formula det(z − L) = det(I − Wz ) det(z − A)

(2.2)

where Wz is the r-by-r matrix given by Wz = Rz xρ , ξσ ,

ρ, σ = 1, . . . , r;

Rz = (zI − A)−1 .

(2.3)

This formula has been extended to infinite-dimensional vector spaces (see Kato [7]); the right-hand side is called the Weinstein – Aronszajn determinant. We specialize the above to rank two and x1 = x,

x2 = y;

ξ1 = ax + by,

ξ2 = cx + dy,

so that L = L(x, y) = A + ax ⊗ x + bx ⊗ y + cy ⊗ x + dy ⊗ y,

(2.4)

where a, b, c, d are constants with determinant ∆ = ad−bc = 0 and x, y linearly independent vectors of V . This defines a 2n-dimensional family of matrices in whose spectra we are interested. In particular the n-dimensional foliation given by isospectral matrices will be of interest to us. The main result of this section is the observation that the eigenvalues of these n  matrices are «in involution» with respect to the symplectic structure dyj ∧dxj , 1

i. e. the natural symplectic structure of T ∗ V = V ∗ × V ∼ V × V . For any two functions F = F (x, y), G = G(x, y) in C 1 (V × V ) we define the corresponding Poisson brackets {F, G} = Fx , Gy  − Fy , Gx , where Fx , Fy are defined by the relation dF = Fx , dx + Fy , dy.

One calls a family of functions F «in involution» if for any two of the F , G ∈ F, one has {F, G} = 0.

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Such a family can be extended by closure under composition: If ϕ, ψ∈ C 1 (R2 , R) then ∂(ϕ, ψ) {ϕ(F, G), ψ(F, G)} = {F, G}, ∂(F, G) i. e., if F , G are in involution, so are ϕ(F, G), ψ(F, G). Instead of showing the involutary character of the eigenvalues, we will establish this for the symmetric functions of the eigenvalues which are rational in x, y. If we apply the formulae (2.2), (2.3) to the case (2.4), we obtain a two-by-two matrix    Qz (x) Qz (x, y) a c , (2.5) Wz = Qz (x, y) Qz (y) b d where Qz (x, y) = Rz x, y,

Qz (x) = Qz (x, x),

and (2.2) becomes det(z − L) = 1 − tr Wz + det Wz = 1 − Φz , det(z − A) where

Φz (x, y) = aQz (w) + (b + c)Qz (x, y) + dQz (y)− − (ad − bc)(Qz (x)Qz (y) − Q2z (x, y)).

(2.6)

(2.7)

Thus the eigenvalues of L are the values of z for which the rational function Φz takes the value 1. If these eigenvalues λj are distinct, the isospectral manifold of matrices (2.4) with spectrum λ1 , λ2 , . . . , λn is given by {x, y | Φλj (x, y) = 1 for j = 1, . . . , n}, and hence it is an algebraic manifold. b. Isospectral Deformations Theorem 1. For any z, z ′ in the resolvent set of A one has {Φz , Φz′ } = 0, i. e. the functions Φz (x, y) are in involution.

(2.8)

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This theorem can be verified by a direct but lengthy calculation. We do not present this, since the result will be a consequence of Theorem 2. Clearly this theorem remains valid if multiple eigenvalues of A are permitted. We extend the class of functions (Φz ) by forming for any polynomial f (z) H(x, y) = 1 f (z)Φz (x, y) dz, 4πi |z|=R

where the circle |z| = R contains the spectrum of A. We express this function explicitly by introducing a basis in which A = diag(α1 , α2 , . . . , αn ) and set f (αj ) = βj ,

β = diag(β1 , β2 , . . . , βn ).

Then one finds 2H = a βx, x + (b + c) βx, y + d βy, y− ad − bc  βi − βj − (x y − xj yi )2 . αi − αj i j 2

(2.9)

i=j

For example, for βi = δik this becomes Gk (x, y) = ax2k + (b + c)xk yk + dyk2 − ′ (xi yk − xk yi )2 − (ad − bc) , αk − αi

(2.10)

i

where the prime indicates that i = k. These functions are all in involution, as a consequence of Theorem 1, and the Φz are recovered by Φz (x, y) =

n  Gj (x, y) , z − αj j=1

and

n

 f (αj )Gj (x, y). H(x, y) = 1 2 j=1

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If (a, b + c, d) = (0, 0, 0), then dGj are linearly independent on an open dense set, while for a = b + c = d = 0 we have the relation n 

Gj = 0.

j=1

In this case we have n − 1 independent commuting functions in |x|2 : G2 , G3 , . . . , Gn , for example. For any choice of the constants f (αj ) = βj the vector field x˙ = ∂ H, ∂y

y˙ = − ∂ H ∂x

(2.11)

is integrable, since G1 , G2 , . . . , Gn are integrals in involution. Hence the spectrum of the matrix (2.4) is fixed under any of these flows, so that in the case of distinct eigenvalues there exists a nonsingular matrix U = U (t) such that U −1 LU is a constant matrix. The infinitesimal version of this statement is that the differential equation (2.11) can be written in the Lax form d L = [B, L] dt

(2.12)

with some matrix B. This is the content of Theorem 2. The vector field (2.11) with H given by (2.9) defines an isospectral deformation of the matrix (2.4) given by (2.12) where β −β  i j (xi yj − xj yi ) . B = 1 (b − c)β + (ad − bc) αi − αj 2

(2.13)

The diagonal elements of the last matrix are zero. Corollary. If H = H(G1 , G2 , . . . , Gn ), then the vector field XH corresponds to the isospectral deformation L˙ = [B, L], where B is of the form (2.13) with βj = 2∂H/∂Gj . Indeed XH =

n  ∂H X = 1  β X , Gj j Gj 2 ∂Gj j=1

where for fixed Gj = cj the βj can be considered as constants. For this vector field Theorem 2 gives the statement of the corollary.

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We show that Theorem 1 is a consequence of Theorem 2. From the form (2.12) of the differential equation (2.11) it is plain that the eigenvalues of L, a hence any function of the eigenvalues are constant along orbits. Therefore  (z − λj ) 1 − Φz =  (z − αj ) is a constant of the motion, for any z, or

d Φ = {Φ , H} = 0. z z dt Taking βj = 2δjk , we get H = Gk ; hence {Φz , Gk } = 0, and hence {Φz , Φz′ } =



(z ′ − αk )−1 {Φz , Gk } = 0.

The proof of Theorem 2 consists of a calculation, which we break into several steps. Setting s = 1 (b + c), 2

r = 1 (b − c), 2

we break L into symmetric and antisymmetric parts: L = A + S + R, S = ax ⊗ x + s(x ⊗ y + y ⊗ x) + dy ⊗ y,

R = r(x ⊗ y − y ⊗ x).

With the determinant ∆ = ad − bc = ad − s2 + r2 , we set β −β  i j B = rβ + ∆Γ, Γ = (xi yj − xj yi ) , αi − αj

the diagonal terms of Γ being zero. The Hamiltonian H is broken up into its quadratic and its quartic part: H = F − ∆G,

F = 1 a βx, x + s βx, y + 1 d βy, y, 2 2  β − β i j (x y − xj yi )2 . G= 1 αi − αj i j 2 i 0,

and to the energy manifold Φ0 (x, y) (see Section 5(c)). The relations Φ0 = 0, Q0 (x, y) = 0 are equivalent to Q0 (x) + 1 = 0,

Q0 (x, y) = 0,

if Q0 (y) = 0

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147

which describes the tangent bundle of the ellipsoid, where the constrained flow takes place. To establish the geodesic flow as an integrable one it suffices to show that the extended flow (3.4) is integrable (see Section 5). This follows from Theorem 1 and the formula n  Gj ; Φz (x, y) = z − αj j=1

hence G1 , G2 , . . . , Gn are integrals of (3.4) which are in involution. b. Isospectral Deformation

Since the zeros of 1 − Φz (x, y) are the eigenvalues of the matrix L(x, y) = A + x ⊗ y − y ⊗ x − y ⊗ y, they also can be viewed as integrals of the motion, and L(x, y) remains similar to itself along the orbits of (3.4). Since the tangents of Uz are given by Φz (x, y) = = 0, it is more natural to construct a matrix L(x, y) whose eigenvalues are the zeros of Φz (x, y). Such a matrix is L(x, y) = Py (A − x ⊗ x)Py where Py = I −

(3.5)

y⊗y

y, y

is for |y| = 1 the projection into the orthogonal complement of y. To see this we observe that the eigenvalue of L(x, νy) = A + ν(x ⊗ y − y ⊗ x) − ν 2 y ⊗ y are given by Φz (x, νy) = 1, or Φz (x, y) = 12 . ν This suggests letting ν tend to infinity. Of course, L(x, νy) has no limit; in fact one of its eigenvalues λ = ν 2 (|y|2 + O(ν −2 )) tends to infinity, and the corresponding eigenvector ϕ = y + ν −1 Py x + O(ν −2 )

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tends to y. Now, if we take ν purely imaginary, then L(x, νy) is Hermitian, and on the orthogonal complement of ϕ this matrix becomes L(x, νy) −

ϕ⊗ϕ , |ϕ|2

which has a limit for ν → ∞ the matrix (3.5). But it is also easy to verify directly that for the matrix (3.5) one has |y|2

det(z − L) = −zΦz (x, y). det(z − A)

(3.6)

Thus the n−1 roots of Φz (x, y) are eigenvalues of L and the nth eigenvalue is λ = 0, which corresponds to the eigenvector y. It is clear, then, that the matrix (3.5) undergoes an isospectral deformation under the flow (3.4). We make this more explicit by writing the differential equations (3.4) in the Lax form d L = [B, L] dt with an appropriate matrix B. We generalize the setup right away and replace the Hamiltonian Φ0 by n    βi − β j βj Gj H= 1 (xi yj − xj yi )2 = 1 βj yj2 + 1 αi − αj 2 2 2

(3.7)

j=1

i 0, the right side of (4.10) is positive and allows for 2n real vectors y. For µ1 = y2 = 1 the µ2 , µ3 , . . . , µn can be viewed as coordinates on the sphere S n−1 which form an orthogonal coordinate system1 . In the following we will derive these properties but ignore the reality conditions. By logarithmic differentiation of (4.10) one finds ⎧ 1 ⎪ ⎪ for k = 1, ⎨ 2µ y ∂y 1 (4.11) = ⎪ ∂µk ⎪ ⎩ 1 (µk − A)−1 y for k  2, 2 which implies

1 E. Rosochatius

& ∂y ∂y ' , = gj δjk , ∂µk ∂µj

refers to them as «elliptische Kugelkoordinaten»: see [16], in particular p. 29.

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with

⎧ 1 ⎪ ⎪ ⎪ ⎨ 4µ

for j = 1,

1

gj =

m′ (µj ) ⎪ ⎪ ⎪ − ⎩ 4a(µj )

(4.12) for j  2.

We verify (4.12) only for distinct µ2 , µ3 , . . . , µn and for µ1 = 0, using the resolvent identity (µk − A)−1 (µj − A)−1 =

−1 ((µj − A)−1 − (µk − A)−1 ). µj − µk

From (4.11) we find for j = k, j, k  2,

( ∂y ∂y ' −1 (Qµj (y) − Qµk (y)) = 0, , = ∂µk ∂µj 4(µj − µk )

and for j = k  2 we compute  & ∂y '2  = 1 (µj − A)−2 y, y = − 1 d Qz (y) , 4 4 dz ∂µj z=µj

which by (4.9) gives the desired result for j  2. For j  2 we have by (4.11) & ∂y ∂y ' = − 1 Qµj (y) = 0, , 4µ1 ∂µ1 ∂µj & ∂y '2 = 1 2 y2 = 1 , 4µ1 ∂µ1 4µ1 which establishes (4.12). Thus we have n 

dyk2 =

n 

gj dµ2j .

j=1

k=1

Having determined y, we turn to the construction of x, which we represent in the orthogonal frame ∂y/∂µj as x=

n  j=1

Xj

∂y . ∂µj

(4.13)

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Taking the inner product with ∂y/∂µk , we find for the coefficients Xj , using (4.11), (4.12), ⎧ ⎪ ⎪ 1 x, y for k = 1, & ∂y ' ⎨ 2µ1 = gk Xk = x, (4.14) ⎪ ∂µk ⎪ ⎩ 1 Qµk (x, y) for k  2. 2

The terms on the right-hand side can be expressed in terms of λ, µ, thus determining x as a function of λ, µ. To show this we compare the coefficients of z −1 as z → ∞ in (4.6): tr(L − A) = so that

n  j=1

(λj − αj ) = (b + c) x, y = 2s x, y,

(4.15)

n

 (λj − αj ).

x, y = 1 2s j=1

To compute Qµj (x, y) for j  2, we set z = µj in (4.6), (4.5), using Qµj (y) = = 0: l(z) = (b + c)Qz (x, y) − bcQ2z (x, y) 1− a(z) for z = µ2 , µ3 , . . . , µn . This is a quadratic equation for Qµj (x, y). If we set ∆ = −bc, 2r = b − c, we obtain for z = µj , j  2, Qz (x, y) −

# b+c = 1 P (z), 2bc a(z)∆

(4.16)

2

P (z) = (a(z)r − l(z)∆)a(z).

Thus by (4.13), (4.14), (4.15), (4.16) we can express x, y in terms of λ, µ, solving the inverse spectral problem.  To construct (M − K ∩ M)/Γ, we construct the orbits of G = 1 Gj = 2

= s x, y, i. e. of

x′ = sx,

y ′ = −sy.

The function F of (4.1) can be chosen as F = y2 and   F = F  e−2st , t=0

j

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and the cross section is given by F = y2 = 1,

or µ1 = 1.

We can take account of the discrete group generated by τk by parametrizing M′ with uj = x2j , vj = xj yj , wj = yj2 . We observe that these 3n functions are rational symmetric functions of the divisor (p2 , p3 , . . . , pn ), # where pj = (µj , P (µj )) is a point on the Riemann surface of (4.2). Indeed, by (4.10) wj = yj2 is a symmetric polynomial of µ2 , µ3 , . . . , µn . By (4.11), (4.12), (4.13), and (4.14) we have vj = xj yj =

n 

Xk

k=1 2 = µ−1 1 x, yyj −

Using x, y = (1/2s)

n 

∂yj yj = ∂µk n  a(µk ) Qµk (x, y)(µk − αj )−1 yj2 . ′ m (µ ) k k=2

(λj − αj ), (4.16), and (4.10), it is clear that vj is also

j=1

a rational symmetric function of the divisor p2 , . . . , pn . Finally, the same is true for uj = vj2 wj−1 and therefore for any rational function of uj , vj , wj . It is well known (see Neumann [13], Siegel [18]) that all rational symmetric functions of p2 , p3 , . . . , pn can be represented in terms of θ-functions on the Jacobi variety and thus also uj , vj , wj can all be represented in this way, giving rise to a map of g → M′ . For nonspecial divisors (p2 , p3 , . . . , pn ) and fixed λ1 , λ2 , . . . , λn this map is given by (4.10), (4.13). Since uj , vj , wj are sufficient to distinguish points on M′ , this mapping is an isomorphism, showing that M′ ∼ g. c. The Symplectic Structure For the proof of Theorem 4 in case (i), it remains to show that the vector fields XGj have constant coefficients with respect to the variables sj of (4.3). n  dyj ∧ dxj in terms of For this purpose we express the symplectic form ω = j=1

the variables λk , µk . Since the λk as well as the µk , being functions of y alone,

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are in involution, it follows that ω has the form ω=

n 

k, j=1

akj dλk ∧ dµj ,

and since dω = 0 one can — at least locally — find a function S = S(λ, µ) such that 2 akj = ∂ S . ∂λk ∂µj Therefore it suffices to compute this function S. Setting ∂λ S = ∂µ S =

n 

n 

Sλk dλk ,

k=1

Sµj dµj , we write also briefly

j=1

ω = ∂λ ∂µ S.

To compute S we use (4.11) to write the 1-form x, dy =

x, dy =

& j

xk dyk as

1

 ∂y ' dµ1 dµj = 1 x, y µ + 1 Qµj (x, y) dµj , 1 2 2 ∂µj n

x,

n 

j=2

or with (4.15), (4.16), and b + c = 2s, # n    P (µj )  dµ1 s 1 1 +

x, dy = dµj . (λk − αk ) µ + 1 4s 2 bc a(µj )∆ j=2 Since ω = −d x, dy = ∂λ ∂µ S, we read off that µj # P (z) dz , a(z) j=2 n

  −S = 1 (λk − αk ) log µ1 + 1 4s 2∆

(4.17)

the integration being taken along paths on the Riemann surface. Of course, S is determined only up to two additive functions of λk and of µk alone. Instead of the λ1 , λ2 , . . . , λn we use as independent variables their symmetric functions σ1 , σ2 , . . . , σn defined by l(z) = z n + σ1 z n−1 + σ2 z n−2 + · · · + σn ,

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so that ω=

 k, j

Sσk µj dσk ∧ dµj =

n 

k=1

dσk ∧ d(Sσk ).

(4.18)

Thus Sσk are canonically conjugate to the symmetric functions σk . For these Sσk we find µ n j  δk1 ∂S 1 z n−k dz, − = log µ1 − # 4s 4 ∂σk P (z) j=2

which shows that Sσk for k = 2, 3, . . . , n agrees, up to the factor 1 with the 4 Abelian differentials of the first kind (4.3). It remains to rewrite the Hamiltonian vector fields XH in these variables. Setting H = H(x, y) = Ψ(σ, µ),

the differential equation takes the form      T 0 −Sσµ µ˙ Ψµ . = Ψσ σ˙ Sσµ 0 In particular, if the Hamiltonian Ψ depends only on the σ, i. e. Ψµ = 0 and det Sσµ = 0, then the system reduces to dS =Ψ σ σ dt

σ˙ = 0, with the solutions

  σ = σ

t=0

,

  Sσ = S σ 

t=0

+tΨσ .

Thus the Sσk vary linearly on M. Hence if we set sk = 4Sσk for k = 1, 2, . . . , n, then the vector field XH for a Hamiltonian H(x, y) = Ψ(σ) becomes XH = 4

n 

k=1

Since by (4.15)

we have

Ψσk ∂ . ∂sk

   G = s x, y = − 1 σ1 + αj , 2 XG = −2 ∂ , ∂s1

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so that XH = 4

n 

k=2

Ψσk ∂ + 2Ψσ1 XG . ∂sk

We apply this remark to H = Gj , which are functions of the σk alone, to show that XGj have modulo XG constant coefficients with respect to ∂/∂sk . This proves Theorem 4 in case (i). d. Degenerate Case In cases (ii) and (iii) we consider the normal form of Section 2, L = A + r(x ⊗ y − y ⊗ x) + dy ⊗ y,

r = 0,

where d = 1 or d = 0. The spectrum of L is given by the zeros of 1 − Φz , where Φz = dQz (y) − r2 (Qz (x)Qz (y) − Q2z (x, y)).

(4.19)

To describe M and M′ we consider again the auxiliary matrix M = Py APy with the spectrum (0, µ2 , µ3 , . . . , µn ) and set µ1 = x, y. The function m(z) is defined by (4.8). As before, we show that x, y can be expressed in terms of µ1 = x, y and of the spectra of L, M . For this purpose we merely have to modify the expressions for the right-hand sides of (4.13) and (4.14). In case d = 0 we have from (4.19), taking the coefficient of z −1 in the expansion at z = ∞, tr(L − A) =

n  j=1

(λj − αj ) = d y2 ,

so that y2 is a function of the λk , while µ1 = x, y is an independent variable on M. The formulae (4.13), (4.14), and (4.16) are to be replaced by n

 ∂y y− 1 , gj−1 Qµj (x, y) 2 2 ∂µj

y j=2 # # Qz (x, y) = 1 (a(z) − l(z))a(z) = 1 P (z) ra(z) a(z)∆ x=

x, y

for z = µ2 , µ3 , . . . , µn . This latter formula agrees with (4.16), since ∆ = r2 .

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These equations, together with (4.10), again give x, y in terms of λ, µ, solving the inverse problem for d = 1. The mapping of M′ into the Jacobi variety of w2 = P (z) is the same as before. We determine the symplectic structure from

x, dy = µ1

y, dy

y2

n

 −1 Qµj (x, y) dµj = 2 j=2

n $  µ  1 1 1 2 2 log y − log y dµ1 − a−1 (µj ) P (µj )dµj , =d 2 2 2∆ j=2

leading to the symplectic form ω = ∂λ ∂µ S with µ

n j    µ1 1 S= log (λk − αk ) + 2 2∆ j=2

# P (z) dz , a(z)

(4.20)

where P (z) = r2 (a − l)l is a polynomial of degree 2n − 1, and the genus of the curve w2 = P (z) is again n − 1. One branch point of the corresponding Riemann surface is at ∞, and n of them are the eigenvalues λk of L. Finally, in the case (iii) with d = 0 one has n  j=1

(λj − αj ) = d y2 = 0,

so that only n − 1 of the λj , say λ2 , λ3 , . . . , λn , are independent variables. Similarly M has only n − 1 independent eigenvalues µ2 , . . . , µn , while µ1 = 0. As additional variables we could use µ1 = G = 1 x, x and F = x, y and 2 proceed as above. This is in accordance with our reduction at the beginning of this section. Notice that the first term in S in (4.17) and (4.20) reflects the exponential and linear behavior of µ1 in t. e. Limit Cases The isospectral manifold of L = L(x, y) is defined by Φλj (x, y) = 1,

j = 1, 2, . . . , n.

In Section 3 we were interested in the manifold Φλj (x, y) = 0,

j = 1, 2, . . . , n − 1,

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which represented the spectrum of Py (A − x ⊗ x)Py . This manifold can be obtained as a limit case from the former, as was shown in Section 3. We want to indicate how one can determine the hyperelliptic curve for this situation, at least for a = 0, b = −c = −d = 1. In Section 3 we saw that the spectrum of Lν = L(x, νy) = A + ν(x ⊗ y − y ⊗ x) − ν 2 y ⊗ y is given by the equation Φz (x, y) = 12 , ν where Φz (x, y) is the function belonging to L1 . If λj (ν) are the eigenvalues of Lν , we set n  (z − λj (ν)) lν (z) = j=1

and have the hyperelliptic curve w2 = Pν (z) = (r2 a(z) − ∆lν (z))a(z). We assume again (a, b, c, d) = (0, 1, −1, −1). In the limit ν → ∞ one root, say λ1 = λ1 (ν), tends to infinity, and ν −2 lν (z) has a limit, namely a polynomial of degree n − 1 having the eigenvalues of Py (A − x ⊗ x)Py as roots. Let l(0) (z) be the polynomial of degree n − 1 and highest coefficient 1 having these roots. Then we conclude  lν (z)  l(0) (z) Φz (x, y) = 12 1 − →k a(z) a(z) ν with some factor k independent of z. This factor is determined by the asymptotic behavior of Φz for large z as k = (a x, x + 2s x, y + d y, y) = 2G(x, y). Thus Φz (x, y) = 2G(x, y)

l(0) (z) , a(z)

so that zl(0) (z) is the characteristic polynomial of Py (A − x ⊗ x)Py .

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We will not repeat the above construction in this case, but note that as auxiliary variables one could use the eigenvalues of either M = Py APy

or N = A − x ⊗ x.

The eigenvalues of the latter equation represent the elliptic coordinates of x, while the eigenvalues of M represent the orthogonal coordinates of y on the sphere |y| = 1 which we used previously. In any event the isospectral manifold M of Py (A − x ⊗ x)Py leads in this case to the Jacobi variety of the hyperelliptic curve w2 = a(z)l(0) (z) = z −1 det(z − A) det(z − L). This case is particularly pleasant, since the branch points of this polynomial are given by the eigenvalues of A and L where the trivial zero eigenvalue of L is omitted. In this case the manifold1 M′ = (M−M∩K)/Γ = M/Γ can be interpreted geometrically. It is the manifold of common tangents to n − 1 confocal quadrics Uλ1 , Uλ2 , . . . , Uλn−1 , where those 2n tangents are to be identified which go into each other by the reflections xj → ±xj . Thus we obtain the result that the common tangents to n − 1 confocal quadrics form a 2n -fold covering of the Jacobian variety (see Staude [20]).

§ 5. Examples of Integrable Flows a. Constrained Systems In the following examples we will have to constrain a Hamiltonian system x˙ = Hy , in the symplectic space (R2n , ω) ω =

m 

y˙ = −Hx

j=1

(5.1)

dyj ∧dxj , to a symplectic submanifold.

For our purposes it will suffice to describe the submanifold M by 2r equations M : F1 (x, y) = . . . = F2r (x, y) = 0. If (5.2) det({Fj , Fk }) = 0 (j, k = 1, 2, . . . , 2r),   then the manifold M is symplectic with ω M the two-form restricted to T M . 1 By

the remarks at the beginning of this section, we can drop K.

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The vector field (5.1), which we will denote also by XH , need not be tangential to M , but we can construct such a vector field by replacing H with HM the restriction of H to M . Then the function HM on the symplectic manifold (M, ωM ) defines a vector field XHM which is tangential to M . This vector field XHM will be called the constrained vector field. There is another way to describe this constrained flow: Since ωM is nondegenerate in T M , there exists a complementary space (T M )⊥ of T M in R2n , which is orthogonal to T M with respect to the symplectic structure. Now it is easily seen that XH − XHM ∈ (T M )⊥ . In other words XHM is the projection of XH into T M with respect to the above splitting of R2n . Effectively one can describe the constrained vector field by a Hamiltonian H∗ = H −

2r 

λj Fj (x, y).

(5.3)

j=1

In order for XH ∗ to be tangential to M we have to require that  λj {Fj , Fk } on M, 0 = XH ∗ Fk = {H, Fk } −

which by (5.2) determines the functions λj uniquely on M . Next we turn to a more special situation where F denotes a class of functions in involution in (R2n , ω), and consider a vector field XH constrained to a symplectic manifold M : F1 = F2 = . . . = Fr = 0,

G1 = G2 = . . . = Gr = 0,

(5.4)

where we assume that F1 , F2 , . . . , Fr , H ∈ F, det{Fi , Gj } = 0 (i, j = 1, 2, . . . , r).

(5.5i) (5.5ii)

Clearly this is a special case of the previous situation since (5.5i, ii) imply (5.2) if we set Fj+r = Gj for j = 1, 2, . . . , r. The constrained Hamiltonian (5.3) takes the form r  (λj Fj + µj Gj ). H∗ = H − j=1

Since the Fj and H are in F and hence in involution, we conclude 0=

r  j=1

µj {Gj , Fk }

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and hence µj = 0. Thus the constrained Hamiltonian has the form ∗

H =H−

r 

λj Fj ,

j=1

and the corresponding vector field is given by  λj XFj XH ∗ = XH −

on M . This has the consequence that for any function E ∈ F we have on M XH ∗ E = 0,  i. e., E M is an integral of the constrained vector field. We conclude that in the situation (5.4), (5.5) the constrained flows have the restriction of the functions in F as integrals. It is evident that these functions in the class FM , obtained from F by restriction to M , are in involution. We will apply this simple device of constraining an integrable system to obtain a new integrable system in the following examples1. b. A Mass Point on the Sphere S n−1 : |x| = 1 under the Influence of the Force −Ax (C. Neumann [14]) The differential equation of this system is x ¨ = −Ax + λx, where λ is chosen so that |x| = 1, x, x ˙ = 0. We obtain this system by constraining the Hamiltonian H = 1 Ax, x + 1 (|x|2 |y|2 − x, y2 ) 2 2 to the symplectic submanifold M : F = 1 (|x|2 − 1) = 0, 2

G = x, y = 0.

1 Note added in proof : Other integrable systems have been recognized as constrained systems. See P. Derft, F. Lund and E. Trubowitz, Nonlinear wave equations and constrained harmonic motion, Comm. Math. Phys. 74, 141–188 (1980).

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169

Observe that {F, G} = |x|2 = 1 on M . Moreover, if F is the class of functions generated by Φz (x, y) = Qz (x) + Qz (x)Qz (y) − Q2z (x, y),

then the expansion at z = ∞ takes the form

  |x|2 Φz (x, y) = z + 12 ( Ax, x + (|x|2 |y|2 − x, y2 )) + O 13 . z z Hence both F , H belong to F, and the flow restricted to M is given by H ∗ = H − λF,

The equations have on M the form

λ = {H, G} = Ax, x.

x˙ = Hy∗ = Hy − λFy = |x|2 y = y, or

y˙ = −Hx∗ = −Hx + λFx = −Ax − |y|2 x + λx, x ¨ = −Ax − (|y|2 − λ)x,

which is the desired flow if |y|2 − λ is renamed λ. The functions in F restricted to M are the desired integrals. In this  case  we have a = −1, d = −c = 1, d = 0 hence the symmetric part of ac db is a0 00 of rank 1. If we set n  Gj (x, y) Φz = , z − αj j=1

we easily see that

2H =

n 

αj Gj (x, y),

2F + 1 =

j=1

n 

Gj (x, y),

j=1

which shows explicitly that H, F are functions of the Gj , which are the integrals of this system. These integrals Gj have been found by K. Uhlenbeck (see [21] and Devaney [3]). c. A Mass Point on the Ellipsoid Q0 (x) + 1 = 0 under the Influence of the Force −ax (Jacobi [6]) The motion of a free particle under the influence of the force −ax is described by the Hamiltonian H = 1 (|y|2 + a|x|2 ). 2

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To describe the restricted motion we introduce the class of functions F generated by Φz (x, y) = aQz (x) + Qz (y) + Qz (x)Qz (y) − Q2z (x, y), which are in involution by Section 2. We set F (x, y) = a + Φ0 (x, y) = = (1 + Q0 (x))(a + Q0 (y)) − Q20 (x, y), G(x, y) = Q0 (x, y), and restrict the motion to F (x, y) = 0,

G(x, y) = 0.

Since H, F ∈ F it follows that the restricted flow is described by H ∗ = H − λF, and hence has all functions of F as integrals. It remains to identify this flow with the desired one. For this purpose we note that G(x, y) = 0 and F (x, y) = 0 implies 1 + Q0 (x) = 0 or a + Q0 (y) = 0. We pick a + Q0 (y) = 0. This condition is invariant under the flow, as one verifies. Thus we have 1 + Q0 (x) = 0, Q0 (x, y) = 0, which means that x lies on the ellipsoid and y is tangential to it at x. The differential equation becomes x˙ = Hy∗ = Hy − λFy = y,

y˙ = −Hx∗ = −Hx + λFx = −ax − 2λ(a + Q0 (y))A−1 x,

or x¨ = −ax − 2λ(a + Q0 (y))A−1 x, which is the equation of the constrained motion. Thus this system is integrable  and Gj  are the desired integrals in involution. For a = 0 we obtain the M geodesic flow on the ellipsoid. Notice that the resulting integrals Gj are obtained from those of the Neumann system (Section 5 (b)) by the symplectic map (x, y) → (y, −x).

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d. Geodesic Flow on the Orthogonal Group (Manakov [8], Mischenko [11]) Arnold studied the geodesic flow on SO(n) under a right-invariant metric, say

$ tr(U˙ T GU˙ ) dt

where G is a fixed positive definite symmetric matrix and U ∈ SO(n). If we define the element A = U˙ U −1 in the Lie algebra, the Euler equations become d (GA + AG) = [A2 , G]. dt Setting L = GA + AG + µG2 , B = A + µG with any scalar µ, these equations can be written in the Lax form d L = [B, L]. dt These formulas were derived by Manakov [8] to construct integrals of this system from the characteristic polynomial of L. We should like to consider a 2n-dimensional subsystem of this flow by setting G = diag(g1 , g2 , . . . , gn ), L = x ⊗ y − y ⊗ x + µG2 , B=

xi yj − xj yi + µG. gi + gj

One verifies that this system is compatible and is described by the Hamiltonian  (xi yj − xj yi )2 . H = −1 gi + gj 2 i 0, . . . , cn > 0 are indeed tori T n . Moreover, on such an invariant compact leaf the differential equation becomes linear. Indeed, in the action-angle variables P , Q the differential equations become Q˙ j = HPj ,

P˙j = −HQj = 0

which are solved by Qj (t) = HPj (P )t + Qj (0),

Pj (t) = Pj (0).

Thus the integration of such integrable systems (with compact leaves) becomes trivial, which explains the name. Actually the commuting integrals Fj can be replaced by other integrals Fk = ϕk (F1 , . . . , Fn )

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187

which are still commuting since {Fk , Fl } =

 ∂(ϕk , ϕl ) {Fi , Fj } = 0. ∂(Fi , Fj ) i, j

Therefore integrals themselves are not of primary interest but the foliation defined by the Pfaffian system dF1 = 0,

dF2 = 0,

...,

dFn = 0

is more important. It defines the foliation which has a geometrical meaning. In the following we will frequently make use of this freedom of replacing a set of integrals by functions of these. 3. Perturbation of integrable systems It is important to realize that among the Hamiltonian systems the integrable ones are exceptional. If an integrable system is perturbed slightly the integrals are in general destroyed. This is related to the often quoted «theorem» of Poincar´e’s about the nonexistence of integrals. In 1923 E. Fermi [8] extended Poincar´e argument to show that on a fixed energy surface there would be no other integrals after general perturbations. However, his argument is only formal and uses the erroneous assumption that for a smooth Hamiltonian system the invariant sets are smooth also. Even though the integrals may get lost after perturbation, the whole foliation does not get lost, but a large subset of the tori (namely the non-resonant tori) do survive small perturbations and form a complicated invariant Cantor set of positive measure. This is the content of the so-called K. A. M. theory which establishes the existence of this set of invariant tori. In a recent paper J. P¨oschel [26] showed that actually the perturbed system can still be viewed as an integrable one when restricted to this Cantor set. This means that one can define n integrals in involution on this Cantor set which are differentiable in the sense of Whitney. However, we will not study such perturbed systems and consider Hamiltonian systems which are integrable in an open set Ω of R2n . Even if a system is integrable it is frequently not easy to find a set of integrals. As a typical example we will discuss in the next section the geodesic flow on an ellipsoid. Here we mention an example which is equivalent to the Kepler problem in Rn , and is given by the Hamiltonian H = 1 |p|2 − |q|−1 . 2

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This system is clearly rotation symmetrical and therefore has many integrals, namely pi qj − pj qi for 1  i < j  n. But these are not commuting integrals, and it takes some experimenting to find the commuting integrals Fk =



1 0 which has a positive imaginary part in the bands when approaching from above. In tho gaps this function is then real. If we form " b(λ) −2 − G(x, x; λ) = Γ(x, λ) a(λ) we obtain a function which is one-valued in the complex plane, has simple poles at αj and is regular at βj . We chose the factor −2 so that this function behaves like λ−1 at λ → −∞ (see (5.5)). Therefore this function is a rational function admitting a partial fraction expansion . n  rj (x) Γ(x, λ) = −2 − ab G(x, x; λ) = . (5.7) λ − αj j=1

# Moreover, since both −b/a and G have a positive imaginary part in the bands, this function is positive in the bands [αj , βj ], j = 1, 2, . . . , n − 1 and [αn , ∞]. This implies rj (x) > 0 j = 1, 2, . . . , n. (5.8)

Proposition 5.1. If G(x, y; λ) is the Green’s function for the potential q and if G satisfies (5.7) then there exists a real solution ψj of Lψj = −ψj′′ + qψj = αj ψj with rj = ψj2 . PROOF. If we represent the solutions ψ+ , ψ− of the previous section as linear combinations of two normalized solutions ϕ1 = ϕ1 (x, λ), ϕ2 = ϕ2 (x, λ) of (L − λ)ϕ = 0 say normalized by     1 0 ϕ1 ϕ2 = ϕ′1 ϕ′2 x=0 0 1 then ϕ1 , ϕ2 are entire functions in λ and from G(x, x; λ) =

ψ+ ψ− [ψ+ , ψ− ]

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221

it follows that rj = Aj ϕ21 + 2Bj ϕ1 ϕ2 + Cj ϕ22

(5.9)

where λ is to be replaced by αj . We have to show that Aj Cj − Bj2 = 0.

(5.10)

But this follows from the differential equation (5.4). For Γ(x, λ) this equation takes the form (5.11) 2(Γ′′ − 2(q − λ)Γ)Γ − Γ′2 = 4 ab . Inserting the expression (5.7) for Γ and using that ϕ = ϕ1 , ϕ2 satisfy (L − αj )ϕ = 0 we obtain as coefficients of (λ − αj )−2 (dropping the index j) 4(Aϕ′1 2 + 2Bϕ′1 ϕ′2 + Cϕ′2 2 )(Aϕ21 + 2Bϕ1 ϕ2 + Cϕ22 )− −4(Aϕ1 ϕ′2 + B(ϕ1 ϕ′2 + ϕ′1 ϕ2 ) + Cϕ2 ϕ′2 )2 =

= 4(AC − B 2 )(ϕ1 ϕ′2 − ϕ′1 ϕ2 )2 = 4(AC − B 2 ). Since the right hand side of (5.11) has only a simple pole at λ = αj it follows indeed (5.10). Hence, if Aj = 0 we can write 2 rj = A−1 j (Aj ϕ1 + Bj ϕ2 ) ,

and because of (5.8) we have Aj > 0. Therefore we can take −1/2

ψj = Aj

Aj ϕ1 + Bj ϕ2 ).

If Aj = 0 then also Bj = 0 and Cj > 0. In this case we can take 1/2

ψj = Cj

ϕ2

proving the proposition. 3. Connection with the mechanical problem From Proposition 5.1 we have the important representation for the function Γ related by (5.7) to the Green’s function: Γ(x, λ) =

n  ψj2 (x) j=1

λ − αj

.

(5.12)

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Since Γ ∼ λ−1 for λ → ∞ we conclude that n 

ψj2 = 1.

(5.13)

j=1

Moreover, ψj are solutions of the equation (L − αj )ψj = 0 which we write in the form ψj′′ = −αj ψj + qψj . (5.14)

The last two equations show that we can interpret xj = ψj (t) as the components of a vector x satisfying the differential equation x ¨ = −Ax + q(t)x,

A = diag(α1 , . . . , αn )

(5.14′ )

where x is restricted to the unit sphere, i. e. ψj (t) are the components of a solution of the Neumann problem (3.11). We recall that this mechanical problem possesses the integrals (see (3.7)) Φλ (ψ ′ , ψ) = (1 + Qλ (ψ ′ ))Qλ (ψ) − Q2λ (ψ, ψ ′ ) where Qλ (ψ) =

n  j=1

hence Qλ (ψ, ψ ′ ) =

ψj2 λ − αj

n  ψj ψj′ j=1

λ − αj

(5.14′′ )

= Γ,

= 1 Γ′ . 2

Another differentiation gives with the aid of (5.13) 1 + Qλ (ψ ′ ) = 1 (Γ′′ − 2(q − λ)Γ). 2 Therefore we find from the differential equation for G, or its equivalent form (5.11) for Γ, Φλ (ψ ′ , ψ) = 1 {2(Γ′′ − 2(q − λ))Γ − Γ′2 } = ab . 4 Hence the solution x = (ψ1 , ψ2 , . . . , ψn ) of Proposition (5.1) corresponds to the solution of the mechanical problem satisfying Φλ (ψ ′ , ψ) =

b(λ) . a(λ)

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In other words, the solution ψ lies on the invariant manifold defined by Φβj (ψ ′ , ψ) = 0, j = 0, 2, . . . , n − 1. Theorem 5.2. If G(x, y; λ) is the Green’s function of the potential q and if G(x, x; λ) satisfies the assumptions (5.2), (5.3) then it can be represented in the form " n 2 a(λ)  ψj (x) (5.15) G(x, x; λ) = − 1 − 2 b(λ) j=1 λ − αj where ψ = (ψ1 , ψ2 , . . . , ψn ) is a solution of the Neumann problem (5.14) satisfying b(λ) Φλ (ψ ′ , ψ) = . a(λ) Moreover, the potential q = q(x) is given by q(x) = 2

n 

αj ψj2 +

j=1

n−1  k=1

βk −

n 

αj .

(5.16)

j=1

It suffices to verify (5.16). For this purpose we recall the asymptotic expansion (4.16) where G1 = q/2. Comparing this with the expansion of the right hand side of (5.15) we find (5.16). 4. Solution of the inverse problem Our problem will be solved if we show the converse to Theorem 5.2, i. e. Theorem 5.3. If ψj is any solution of the problem (5.14) with Φλ (ψ ′ , ψ) =

b(λ) a(λ)

(5.17)

then the function q(x) defined by (5.16) is the potential of an operator (4.1) with the band spectrum (5.1). We indicate the proof: With the given ψj and the q defined by (5.16) we have (L − αj )ψj = 0.

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But we need the solutions of (L − λ)ψ = 0

(5.18)

for arbitrary complex λ. For this purpose we define G(x, λ) = G(x, x; λ) by (5.15). Then the relation (5.17) is equivalent to the differential equation (5.4). It also clearly satisfies (5.5) and therefore G(x, λ) is the candidate for the Green’s function on the diagonal. Moreover, we show that Im G(x, λ) > 0

for Im λ > 0.

(5.19)

By the maximum principle for harmonic functions it suffices to show Im G(x, λ)  0 on the real axis. In the gaps one has Im G(x, λ) = 0 by (5.15) since both factors are real. In the bands, however, G(x, λ) is purely imaginary and Im G(x, λ) > 0 for αj < λ < αj + ε and ε > 0 small. It suffices to show that G has no zero in a band. This follows from (5.4) which gives G′ = ±1 at a zero of G, hence G would be real near such a zero, while actually it is purely imaginary. Hence the zeros of G must lie in the gaps. In order to determine the spectrum of L we have to find the Green’s function G(x, y; λ) of L, since we have not yet seen that G(x, λ) is related to G(x, y; λ) on the diagonal. For this we first show that for Im λ > 0, ϕ(x, λ) =

#

  x G(x, λ) exp − 1 G−1 (t, λ) dt 2

(5.20)

0

is a solution of (5.18). Note that the integrand is well defined because of (5.19). To verify this claim we form ϕ′ 1 G′ − 1 1 −1 1 G′ ϕ = 2 G − 2G = 2 G and

 ϕ′ ′  ϕ′ 2 ϕ′′ = 1 2 {2GG′′ − G′2 + 1}. = ϕ ϕ + ϕ 4G Because of (5.4) this agrees with ϕ′′ 1 2 ϕ = 4G2 4(q − λ)G = q − λ, proving our claim.

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If we could show that for Im λ = 0 −1 lim x→∞ 2x

x

G−1 (t, λ) dt = w(λ)

0

satisfies Re w(λ) < 0 then the above solution (5.20) belongs to L2 (0, ∞) and can be used for ψ+ (x, λ). Similarly we find  x  # ψ− (x, λ) = G(x, λ) exp 1 G−1 (t, λ) dt 2 0

and therefore   x # 1 −1 G(x, y; λ) = G(x, λ)G(y, λ) exp − G (t, λ) dt 2

for x > y.

y

In particular, we see that G(x, x; λ) agrees with the function given by (5.15). From the fact that G(x, λ) is real in the gaps and purely imaginary and = 0 in the bands one verifies that the spectrum consists of the bands. To prove the inequality Re w(λ) < 0 we use the fact that the potential q so constructed is almost periodic, and therefore the solutions ψ+ , ψ− decay at an exponential rate for x → +∞ or x → −∞. This follows from (4.15). From this it follows that the mean value w(λ) of the exponent in (5.20) has a non vanishing real part. Since Re w(λ) < 0 for λ → −∞ we have Re w(λ) < 0 for Im λ = 0. This completes the proof of Theorem 5.3. 5. Finite gap potentials as almost periodic functions In Section 3 we showed that the solutions of the Neumann problem are quasi-periodic functions with generally n − 1 = N basic frequencies. Therefore also the potentials q defined by (5.16) are quasi-periodic. It is natural to study the Floquet exponent w(λ) and α(λ) = Im w(λ) as defined in Section 4 for these potentials. Theorem 5.4. For the above potentials q defined by (5.16) p(λ) dλ p(λ) dλ = # dw = # 2 −a(λ)b(λ) 2 −R(λ)

(5.21)

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is a differential of the third kind, where p(λ) = λn−1 + . . . is a polynomial of degree n − 1. This follows immediately from the formula (4.14) dw = − 1 2 dλ

. . n  M (ψj2 ) p = −1 −a a −a 2 b λ − αj b j=1

where p is a polynomial of degree n − 1 with highest coefficient n 

M (ψj2 ) = 1

j=1

(see (5.13)). Secondly we study α(λ) = Im w(λ) in the bands. We can write G(x, λ) = = iK(x, λ) with K(x, λ) > 0 in the interior of the band. Hence   w(λ) = M − 1 = i M (K −1 ) 2 2G w′ (λ) = M (G) = iM (K) and for α(λ) = Im w(λ) we obtain 2α dα = M (K −1 )M (K) > M (1) = 1 dλ by Schwarz’ inequality. We have equality only if K = const, i. e. G′ = 0 which implies q = const. Hence we have Theorem 5.5. In the bands we have the inequality d(α2 ) > 1. dλ This inequality is, of course, not only applicable in the finite band case, but in all intervals of the real axis where G is purely imaginary, for example, in the periodic case, where we used it to estimate the band length.

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6. The elliptic coordinates on the sphere At the end of Section 3 we introduced the elliptic coordinates µk on the sphere; we shall interpret them here for the spectral problem and express the potential q in terms of them. These coordinates µk were introduced as zeros of

Qλ (ψ) =

n  ψj2 (x) j=1

λ − αj

=

n−1  k=1

(λ − µk ) a(λ)

.

(5.22)

Because of (5.15) they agree with the zeroes of the Green’s function on the diagonal G(x, x; λ). They depend on x and are restricted to the gaps. Indeed we showed that G(x, x; λ) = 0 in the bands. Moreover, from (5.22) it is clear that each interval (αj , αj+1 ), j = 1, 2, . . . , n − 1 contains one simple zero. Therefore we conclude that each gap (βj , αj+1 ) j = 1, 2, , . . . , n − 1, contains one zero, or βj  µj (x)  αj+1 . The potential q = q(x) can be expressed in terms of these variables µj (x) by the formula n−1  (αk+1 + βk − 2µk ), (5.23) q(x) − α1 = k=1

which was derived by McKean and Trubowitz [18] in the periodic case, also in the presence of infinitely many gaps. For the proof we compare the coefficients of λ−2 in the expansion of (5.22) at λ → ∞ to get n−1 n n    αj − µk . αj ψj2 = j=1

j=1

k=1

If we insert this into (5.16) we obtain formula (5.23). Note that the terms on the right hand side of (5.23) lie in the interval [−αk+1 + βk , αk+1 − βk ] so that we obtain simple bounds for q(x) − α1 : |q(x) − α1 |  in terms of the total gap length.

n−1  k=1

|αk+1 − βk |

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7. Alternative choice of the branch points In the above consideration the choice of the αj , βj is rather arbitrary. What matters are the branch points λ0 , λ1 , . . . , λ2N of the Riemann surface Z=

#

−R(λ);

R(λ) =

2N 

j=0

(λ − λj ).

The factorization R(λ) = a(λ)b(λ) =

n 

(λ − αj )

j=1

n−1  k=1

(λ − βk )

corresponding to (5.6) could be replaced by any other factorization R(λ) = a∗ (λ)b∗ (λ) =

n 

(λ − α∗j )

j=1

n−1  k=1

(λ − βk∗ ).

Since the Green’s function is independent of this factorization we obtain analogous to (5.15), " n ∗2 a∗ (λ)  ψj G(x, x; λ) = − 1 − ∗ 2 b (λ) j=1 λ − α∗j with solutions ψj∗ (x) of

(ψj∗ )′′ = (q − α∗j )ψj∗ .

This gives a large number of quadratic relations between these solutions ψj ; ψj∗ at the various branch points. This freedom of choice can be used to study those non-degenerate orbits of the mechanical problem which we neglected so far. Until now we restricted ourselves to the case where α1 < β1 < α2 < . . . < βn−1 < αn which is equivalent to the condition (see (5.17)) Φλ (ψ ′ , ψ) =

n  Fj (ψ ′ , ψ) b(λ) , = λ − αj a(λ) j=1

Fj (ψ ′ , ψ) > 0 (j = 1, 2, . . . , n).

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If we have a different arrangement of the distinct αj , βk we can always reduce it to the above case by picking the α∗j , βj∗ interlacing and making use of the above identities.

Fig. 1

§ 6. Limit Cases, Bargmann Potentials 1. Schwarz – Christoffel mapping The Floquet exponent w = w(λ) for the finite band case gives rise to a conformal mapping of Im λ > 0 into a slit domain. From (4.15) it is clear that the image domain lies in the second quadrant. Moreover, α(λ) = Im w(λ) is constant in the gaps (see theorem 4.7). Since α(λ) = 0 for real λ  λ0 = α1 this part of the real axis is mapped on the negative real axis. If one uses that G(x, x; λ) = 0 and purely imaginary on the bands one verifies easily that w = = w(λ) maps the upper half plane into the slit domain shown in the figure. The n − 1 gaps of finite length go into n − 1 slits, and the bands are mapped onto parts of the imaginary axis.

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This mapping is given by the Schwarz – Christoffel formula

w(λ) =

λ

λ0

−1/2   2N (λ − λj ) dλ p(λ) − j=0

where p(λ) is a polynomial of degree n − 1 = N . This is in agreement with (5.21). The zeros of p(λ) √ are mapped into the tips of the slits. From the asymptotic behavior w ∼ − −λ one sees that this mapping is bijective. Instead of prescribing λ0 , λ1 , . . . , λ2N we could equally well prescribe the slit domain characterized by ωj , hj > 0, j = 1, 2, . . . , N = n − 1 and λ0 = α1 , where iωj corresponds to w(αj+1 ) = w(βj ) and hj > 0 to the «height» of the j-th slit. By the Riemann mapping theorem there is a unique conformal mapping of the slit domain characterized by ωj , hj (j = 1, 2, . . . , n − 1), onto the upper half plane, taking w = 0 into λ = λ0 = α1 and with the asymptotic behavior √ λ ∼ −w2 or w ∼ − −λ at infinity. Clearly λ0 = α1 can be normalized to be zero by a translation, so that the spectrum is characterized by the 2N positive numbers λj − λ0 (j = = 1, 2, . . . , N ) or equivalently by the 2N positive numbers ωj , hj . 2. Basis for the frequency module



We know from the study of the mechanical model that the finite band potentials are quasi-periodic and the frequency module (q) is spanned in general by n − 1 = N frequencies. Theorem 6.1. For a finite band potential q = q(x) the frequency module is spanned by ωj = w(βj ), j = 1, 2, . . . , n = N − 1. The proof turns out to be relatively simple if we use the connection with the mechanical problem. ′ (q), where we assume at be a frequency basis of Let ω1′ , ω2′ , . . . , ωN first that we are in the generic situation of N rationally independent frequencies. Then we know from Theorem 4.7 that



2iωk = 2w(βk ) = i

N 

ν=1

jνk ων′

k = 1, 2, . . . , N

(6.1)

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with integers jνk . Of course, the ων′ are determined only up to a unimodular transformation. To determine these integers we will deform the heights hk → 0 and verify that in that case a frequency basis is approximately given by # (6.2) ων′ ∼ 2 αν+1 − α1 (ν = 1, 2, . . . , n − 1) # (6.3) ων ∼ αν+1 − α1

so that in this limit case we can take jνk = δνk in (6.1). Because of the continuous dependence of the frequencies on the hk and the fact that the jνk are integers, we conclude that ωk′ = ωk form a frequency basis also for positive hk . By another continuity argument we can free ourselves from the assumption that the ων′ are rationally independent. It remains to prove (6.2) and (6.3). The latter is simple enough, since for hk → 0 the potential tends to the constant potential q(x) → α1 and therefore w(λ) → −(α1 − λ)1/2 . Hence iων = w(αν+1 ) → −(α1 − αν+1 )1/2 = i

# αν+1 − α1 ,

proving (6.3). In order to verify (6.2) we have to find a frequency basis for q(x) if hk are small, or if the n − 1 gap lengths αν+1 − βν tend to zero. For this purpose we locate the solutions of the mechanical problem (5.14) which correspond to αν+1 − βν = 0. Because of (5.17) we have in this limit situation n  Fj (ψ ′ , ψ) b(λ) 1 ′ = = Φλ (ψ , ψ) = λ − αj λ − α1 a(λ) j=1

since the other zeros and poles cancel pairwise. Thus we have for these solutions F1 = 1,

F2 = F3 = . . . = Fn = 0.

Using the explicit form (3.9) of the Fk we sec that because of α1 < < α2 < . . . < αn all terms of Fn are positive, hence Fn = 0 implies ψn = = 0. Inductively, we conclude ψ2 = ψ3 = . . . = ψn = 0, ψ1 = ±1. Hence the vector ψ = ψ(x) corresponds to the stationary solution ψ = ±e1 of (5.14).

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To study the solutions near this equilibrium solution we linearize (5.14). Using δνj = ξj , j = 2, . . . , n as local coordinates near ψ = ±e1 we find from (5.14) ξ¨j = (−αj + α1 )ξj , j = 2, . . . , n. √ The characteristic exponents are purely imaginary, namely ±i αj − α1 (j = = 2, . . . , n), and the stationary solution is stable. The solutions are linear combinations of # exp(±i αj − α1 x), √ with frequencies αj − α1 . On account of the formula (5.16) the frequencies √ of q are given by ωj′ = 2 αj − α1 which proves (6.2) and hence Theorem 6.1. As a consequence of Theorem 6.1 we see that a finite gap potential of period l = π is characterized by integers ωj . Actually this holds true even if one has infinitely many gaps. More generally, this theorem allows us to construct finite gap potentials with prescribed frequency module. 3. Stationary solutions and their stability behavior We saw that ±e1 are stationary solutions of the mechanical problem. It is easily seen that the most general stationary solutions are given by the eigenvectors of A = diag(α1 , α2 , . . . , αn ), i. e. ψ = ±ek . To study their stability behavior we compute the 2(n − 1) characteristic exponents, which are √ ± αk − αν ;

ν = k.

(6.4)

Therefore 2(k − 1) of these exponents are real and the remaining 2(n − k) are purely imaginary. Hence ±e1 are the only stable stationary solutions and ±ek for k  2 has an unstable manifold W− (±ek ) of dimension k − 1, and a stable manifold W+ (±ek ) of the dimension n − k. Now it is clear that the solutions on the stable or unstable manifold approach the stationary solution at an exponential rate for t → +∞, t → −∞, respectively. In any event, these solutions are not quasi-periodic and we want to investigate their meaning for the spectral problem. For this purpose we observe that for ψ = ±ek , ψ ′ = 0 the integrals Fj (see (3.9)) satisfy  Fj (ψ ′ , ψ) = 0 for j = k (6.5) Fk (ψ ′ , ψ) = 1

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and hence with (3.8) Φλ (ψ ′ , ψ) =

1 . λ − αk

(6.6)

Comparing this with (5.17) we can view this as the limit case where βν − αν → 0 αν+1 − βν → 0

for ν = 1, 2, . . . , k − 1 for ν = k, . . . , n.

In other words, we collapse the first k − 1 bands and the last n − k gaps. In the limit we have a half infinite slit from αk to +∞ and k − 1 points αν = βν for ν  k − 1. ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ s♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ s♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣s♣ α1 = β1 αk−1 = βk−1 αk This corresponds to a Riemann surface which is a sphere with k − 1 punctures. The differentials of the first kind can be integrated in terms of logarithms in this case and we will be able to write down the corresponding solutions and potentials explicitly. The unstable and stable manifolds W± (ek ) clearly also lie on the integral manifolds given by (6.5) or equivalently by (6.6). From the explicit formula (3.9) we conclude from (6.5) similarly as before that on W± (ek ) one has ψk+1 = . . . = ′ = ψn = 0, ψk+1 = . . . = ψn′ = 0 and it suffices to study ψ1 , ψ2 , . . . , ψk in its dependence on t. In other words, the problem is reduced to the motion of the mass-point on a (k −1)-dimensional sphere and n is replaced by k. Equivalently, it suffices to consider the case k = n and the motion on W± (±en ). 4. The flow on the unstable manifold W+ (en ) At first it may appear as a surprise that the exponentially decaying solution, the stable manifold, can appear as the limit situation of almost periodic solutions. But this phenomenon is easily illustrated with the example of the nonlinear pendulum given by the differential equation y¨ = κ 2 sin y, which has the integral G=

y˙ 2 y y˙ 2 + κ 2 (cos y − 1) = − 2κ 2 sin2 . 2 2 2

(6.7)

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The familiar phase diagram for the level lines shows that the solutions for G positive or negative are periodic but for G = 0 asymptotic to the unstable equilibrium y = 0 (mod 2π).

Fig. 2

It turns out that this example of the nonlinear pendulum corresponds to the motion of the Neumann problem for n = 2. In fact, in this simple case we have the motion on the circle ψ12 + ψ22 = 1. If we parametrize this circle by the angle θ: ψ1 = sin θ,

ψ2 = cos θ,

the differential equations become 2θ¨ = (α2 − α1 ) sin 2θ, which for y = 2θ, κ 2 = α2 − α1 agrees with (6.7). Then the solutions of this equation are given by elliptic integrals, which degenerate on the stable and unstable manifolds G = 0. In this case we have √ θ˙ = ±κ sin θ, κ = α2 − α1 . We integrate these equations by introducing the independent variable τ = tan θ/2 go that ψ1 = sin θ = 2τ 2 1+τ ψ2 = cos θ =

1 − τ2 1 + τ2

dτ = θ˙ 1 + τ 2 = ±κτ 2 dt

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or τ = τ (0) exp(±κt). Therefore the solutions become up to phase shift and sign change: ψ1 (t) = (cosh κt)−1 ;

ψ2 (t) = tanh κt.

(6.8)

We interpret these solutions for the spectral problem. By (5.16) and α1 = β1 they give rise to a potential q(t) = 2(α1 ψ12 + α2 ψ22 ) − α2 or q(t) − α2 =

− 2κ 2 , (cosh κt)2

κ 2 = α2 − α1 .

(6.9)

This is the wellknown (reflectionless) potential for which the spectrum is continuous in [α2 , ∞], and which has a point eigenvalue at λ = α1 < α2 with eigenfunction ψ1 (t) (sec (6.8)). Note that ψ2 is, of course, a solution of ϕ′′ = = (q − λ)ϕ for λ = α2 , but ψ2 is not in L2 (R), and therefore λ = α2 is no eigenvalue. These formulae can be generalized to arbitrary n and we can find explicit expressions (rational in exponentials) for the solutions on W± (en ). The corresponding potentials are the Bargmann potentials. As a matter of fact, in this case we start with the formulae for the Bargmann potential and derive the solutions for the mechanical problem. 5. The Bargmann potentials We make use of the explicit formula for the reflectionless potentials with pre2 , where 0 < κ1 < κ2 < . . . < κN , scribed negative eigenvalues −κ12 , . . . , −κN which were constructed by V. Bargmann. They can be written in the form ⎧  2 d ⎪ log ∆ q = −2 ⎪ ⎪ ⎪ dx ⎨  ηi ηj  (6.10) ∆ = det δ + i, j = 1, 2, . . . , N ij ⎪ ⎪ κi + κj ⎪ ⎪ ⎩ ηj = aj exp(−κj x).

Here a1 , . . . , aN are N arbitrary constants. These are rational functions of the exponentials ηj = aj exp(−κj x). The original derivation of these formulae was based on the inverse theory of Gelfand and Levitan.

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There is another representation which can be derived rather directly by inserting one eigenvalue at a time. For this purpose we represent the above q in a different form. For details see [5]. Set  cosh(κj x + δj ) for j odd χj = (6.11) for j even sinh(κj x + δj ) and define the Wronskian WN = W (χ1 , χ2 , . . . , χN ). If one writes this determinant as a polynomial in ξj = exp(κj x + δj ) and ξj−1 one finds that the coefficients are positive and hence WN > 0. (It is for this reason that the χj were chosen alternatively as ch, sh). One verifies that WN is related to the determinant ∆ in (6.10) by ∆ = 2N

N   exp(κj x + δj )WN (κi − κj )

where N −j ′

aj = (−1)

(6.12)

j=1

i>j

r (κj ) exp(2δj );

N  z − κi . r(z) = z + κi i=1

Therefore



   d 2 log ∆ = d 2 log W , N dx dx and we can represent q by the formula (6.10) with WN in place of ∆. If we set ϕ(x, k) =

W (χ1 , χ2 , . . . , χN , exp(kx)) WN

(6.13)

where the numerator is the Wronskian of the N + 1 functions in the argument, then  2 d ϕ(x, k) = (q + k 2 )ϕ(x, k); dx in other words, we have explicit formulae for the solution of the differential equation with λ = −k 2 . For k = κj one finds the asymptotic behavior ϕ(x, k) ∼

N 

(k − κj ) exp(kx)

j=1

for x → ±∞.

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But for k = κj these solutions are square integrable and decay for x → ±∞. We find ϕ(x, κj ) ∼ const · exp(−κj x) for x → +∞. (6.14) For k = κj we can simplify the formula (6.13) to ⎧ W (χ1 , . . . , χ !j , . . . , χN ) ⎪ ⎪ ⎪ ⎨ ϕ(x, κj ) = cj WN  ⎪ j ⎪ c = exp(δ )(−1) κ (κj2 − κν2 ) j j j ⎪ ⎩

(6.15)

ν =j

where the carret indicates that χ !j has to be omitted. Now we claim that ⎧ exp(δj ) ⎪ ⎪ ⎪ ψj = bj ψ(x, κj ); bj =  2 ⎪ ⎪ |κj − κν2 |1/2 κj ⎪ ⎨ ν =j

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ0 = b0 ϕ(x, 0);

b0 =

N 

(6.16)

κj−1

j=1

are solutions of the differential equation ψj′′ = (q + κj2 )ψj ,

j = 1, 2, . . . , N

(6.17)

which satisfy the identity N 

ψj2 = 1.

(6.18)

j=0

To prove this we note that ϕ(x, k) ∼ exp(kx) f (x, k) =  (k − κj )

for x → ∞.

We can use f (x, k), f (x, −k) as solutions decaying at x → +∞, x → −∞ respectively. Their Wronskian is −2k, and we look at the function f (x, k)f (x, −k) = k

ϕ(x, k)ϕ(x, −k) N  (k 2 − κj2 ) (−1)N k j=1

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which is rational in k. For k → ∞ it behaves like k −1 and therefore we have f (x, k)f (x, −k) 1= 1 dk 2πi k |k|=R

if we integrate over a large circle. If we compute the residues at k = ±κj and use ϕ(x, −κj ) = (−1)j exp(δj )ϕ(x, κj ) we obtain (6.18). Now we interpret the formulae (6.17), (6.18) as solution of the mechanical problem. We set αn = 0,

αn−j = −κj2 ,

j = 1, 2, . . . , N = n − 1,

and see that the ψj of (6.16) are solutions of the Neumann problem. To be precise the labelling has to be reversed: j → n − j. Because of (6.14) we see that ψj ∼ const · exp(κj x), i. e. these solutions approach the ψj = ±δjn at an exponential rate. In fact, because of # # κj = −αn−j = αn − αn−j ,

they are precisely the characteristic exponents at en , and the solutions (6.16) are solutions on the stable manifold W+ (en ). Since they depend on N = n − 1 parameters δj they represent all solutions of W+ (en ). Thus we found explicit formulae for the solutions on the stable manifold and see that they are rational functions in the exponentials exp(κj x + δj ). For N = = n−1 = 1 these are precisely the solutions (6.8) for δ1 = 0. The corresponding potentials q(x) are also rational in exponentials and decay for x → ±∞. The geometrical picture shows clearly how these decaying Bargmann potentials are obtained as limit cases of quasi-periodic potentials. Geometrically this corresponds to singularities of the foliation given by the integrals, due to the linear dependence of the gradients of the integrals. 6. A focussing property on S 2 For a later application we write down these solutions explicitly for N = 2, i. e. for the flow on the two-dimensional sphere in R2 . In this case one finds χ1 = cosh(κ1 x + δ1 ), W2 =

χ1 χ′2

χ2 = sinh(κ2 x + δ2 ) − χ′1 χ2 > 0

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and

⎧ $ χ2 ⎪ ⎪ ψ1 = − κ22 − κ12 ⎪ ⎪ W 2 ⎪ ⎪ $ ⎪ ⎨ χ1 2 2 ψ2 = κ2 − κ1 W2 ⎪ ⎪ ⎪ ⎪ 2 ′ ⎪ κ χ χ2 − κ12 χ1 χ′2 ⎪ ⎪ ⎩ ψ0 = κ 1κ 2 1 1 2 W2

239

(6.19)

and one verifies that these are solutions of ψj′′ = (q + κj2 )ψj , with q = −2 which satisfy



j = 0, 1, 2; d dx

2

κ0 = 0

log W2

ψ02 + ψ12 + ψ22 = 1.

(6.20)

Moreover, for x → ∞ we find ψ0 → 1,

ψ1 → 0,

ψ2 → 0

which expresses that these solutions belong to the stable manifold W+ (e0 ). It is a remarkable geometrical fact that these orbits of W+ (e0 ) all pass through another point, namely the point .  κ 2   κ 1 . (6.21) (ψ0 , ψ1 , ψ2 ) = − κ , 0, ± 1 − κ1 2 2

Indeed, when an orbit passes the plane ψ1 = 0 then by (6.19) we have χ2 = cosh(κ2 x + δ2 ) = 0 and therefore

κ12 χ1 χ′′2 κ = − κ1 . 2 χ1 χ′ 2 2

ψ0 = − κ 1κ 1

The third component follows from (6.20). For the confocal cones χ20 χ21 χ22 + + =0 λ λ + κ12 λ + κ22

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one obtains a singular cone for λ = −κ12 given by the lines through the points (6.21). The observation that all orbits of the stable manifold W+ (e0 ) pass through the focal point was pointed out to me by R. McGehee. 7. N -solitons It is well known that the formula (6.10) can be used to construct special solutions of the Korteweg – de Vries equation (1.2). More precisely, if we replace in (6.10) ηj by ηj = aj exp(−κj x + 4κj t3 ) then the resulting potential q = q(x, t) is a solution of the KdV equation. Analogously, one has to replace (6.11) by  for j odd cosh θj χj = sinh θj for j even with θj = κj (x − 4κj2 t) + δj . These potentials q(x, t) resolve for t → ±∞ into N solitons and therefore one refers to these solutions as N -solitons. The formulae (5.16) go over into q(x, t) = −2

N 

κj2 ψj2 ,

j=1

where ψj = ψj (x, t) are defined by (6.16) with the indicated replacement. We want to interpret the focusing effect on the two-sphere to show the surprising fact that the graph of the 2-soliton   t, x, q = −2(κ12 ψ12 + κ22 ψ22 ) contains a straight line given by

x − 4κ22 t = const,

q = −2(κ22 − κ12 ).

Hence on the 2-soliton there exists a position which moves with the velocity of the faster soliton having constant elevation. The proof is straightforward. We determine x so that ψ = 0 which by (6.19) amounts to χ2 = sinh θ2 = 0

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241

or δ x − 4κ22 t = − κ2 = const. 2

These values of x correspond to the focus where we have by (6.21)  κ 2 ψ02 = κ1 , 2

ψ1 = 0,

 κ 2 ψ22 = 1 − κ1 2

and therefore q = −2κ12 ψ12 − 2κ22 ψ22 = −2(κ22 − κ12 ) proving our claim. It would be interesting to find a generalization of this result to the N -soliton and an analogue focussing for the mechanical problem on S N . 8. Concluding remarks With these considerations we wanted to show how the interplay between the inverse spectral problem and the mechanical problem on the sphere can be used to advantage. We found the explicit orbits on the stable manifolds starting from the formulae for the Bargmann potentials. These potentials appeared as the limit case of the finite band potentials where all bands collapse to points, the eigenvalues of this potential. But this interplay can be used to study further interesting potentials for which the spectrum can be determined explicitely. For example, in the neighborhood of one of the stationary solutions ek with purely imaginary and real eigenvalues one finds unstable periodic orbits. These periodic orbits in turn have stable and unstable manifolds and they can be described in terms of exponents exp(rj x), rj real, and purely imaginary exponentials, due to the periodic behavior. Therefore the corresponding potential will exponentially approach periodic orbits as x → ±∞. Their spectrum will be given by two bands (one finite one semi-infinite) and point eigenvalues corresponding to the real exponentials. Similarly, one can consider unstable invariant tori, and the corresponding stable and unstable manifolds (whiskers) corresponding to potentials which are asymptotic to quasi-periodic functions as x → ±∞. These are simply limit cases of the quasi-periodic potentials discussed in the proceeding Section. Added in proof : Since those lectures were given the interesting papers B. M. Levitan, Almost periodicity of infinite-zone potentials, Math. USSR, 18 (1982), 249–273;

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B. M. Levitan, Approximation of infinite-zone potentials by finite-zone potentials. Izv. Akad. Nauk USSR, ser. math., 46, №1 (1982). 56–87; appeared. They extend the construction of the finite-band potential to the case of infinitely many bands. In the periodic case this had been done by McKean – Trubowitz [18] but Levitan’s approach is in contrast to [18] based on the study of the Jacobi inversion in the infinite genus case. These papers contain further interesting related references.

References [1] M. Adler, and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras and Curves, Advances in Math., 38 (1980), 267–317. [2] M. Adler, and P. van Moerbeke, Linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory, Advances in Math., 38 (1980), 318–379. [3] S. I. Al’ber, On Stationary Problems for Equations of Korteweg – de Vries Type, Comm. Pure Appl. Math., 34 (1981), 259–272. [4] V. Bargmann, Remarks on the determination of the central field of force from the elastic scattering phase shift, Phys. Rev., 75 (1949), 301–303. [5] P. Deift, and E. Trubowitz, Inverse Scattering on the Line, Comm. Pure Appl. Math., 32 (1979), 121–125. [6] A. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg – de Vries type, finite zone linear operators and Abelian varieties, Russian Math. Surveys, 31 (1976), 59–146. [7] L. Faddeev, The inverse problem in quantum theory of scattering, Uspehi Mat. Nauk, 14 (1959) 57–119. [8] E. Fermi, Beweis, dass ein mechanisches Normalsystem im allgemeinen quasiergodisch ist, Phys. Z., 24 (1923), 261–265. [9] H. Flaschka, The Toda lattice I, Phys. Rev. B, 9 (1974), 1924–1925. [10] C. H Gardner, J. M. Greene, M. D. Kruskal, R. Miura, Korteweg – de Vries equation and generalization VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97–133. [11] M. Henon, Integrals of the Toda lattice, Phys. Rev. B, 9 (1974), 1921–1923.

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References

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[12] R. Johnson, and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403–438. [13] H. Kn¨orrer, Geodesics on Quadrics and a Mechanical Problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69–78. [14] P. D. Lax, Nonlinear partial differential equations of evolution, Proc. Int. Cogr. Math. (Nice 1970), Gauthier-Villars, 831–840, Paris 1970. [15] P. D. Lax, Periodic solutions of the KdV equation, Comm. Pure Appl. Math., 28 (1975), 141–188. [16] S. V. Manakov, Complete integrability and stochasticity for discrete dynamical systems, Journal E. Theor. Phys., 40 (1974), 269–274. [17] H. P. Mc Kean, and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math., 30 (1975), 217–274. [18] H. P. Mc Kean, and E. Trubowitz, Hill’s operator and Hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math., 29 (1976), 143–226. [19] H. P. Mc Kean, Integrable Systems and Algebraic Curves, Lecture Notes in Math., 775, 83–200, Springer-Verlag, 1979. [20] J. Moser, The scattering problem for some particle systems on the line, Lecture Notes in Math., 597, Geometry and Topology, 441–463, Springer-Verlag, 1977. [21] J. Moser, Three integrable Hamiltonian systems, Advances in Math., 16 (1975), 197–220. [22] J. Moser, Varios Aspects of Integrable Hamiltonian Systems, Progress in Math., 8, 233–289, Birkh¨auser, Boston, 1980. [23] J. Moser, Geometry of Quadrics and Spectral Theory, The Chern Symposium 1979, Springer-Verlag, 1980. [24] J. Moser, An example of a Schr¨odinger operator with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv., 56 (1981), 198–224.

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[25] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation. Korteweg – de Vries equation and related nonlinear equations, Proc. Int. Symp. On Algebraic Geometry, Kyoto 1977, 115–153, Tokyo 160–91, Japan. [26] J. P¨oschel, Integrability of Hamiltonian Systems on Cantor Sets, Comm. Pure Appl. Math., 35 (1982), 653–696. [27] G. Scharf, Fastperiodische Potentiale, Helv. Phys. Acta. 24 (1965), 573–605. [28] R. Schrader, High energy behavior for non-relativistic scattering by stationary external metrics and Yang – Mills potentials, Z. Physik C, Particles and Fields, 4 (1980), 27–36. [29] A. P. Veselov, Finite band Potentials and an integrable Systems on the sphere with quadric Potentials, Functional Anal. Appl., 14 (1980), 48–50 (Russian). ¨ [30] H. Weyl, Uber gew¨ohnliche Differentialgleichungen mit Singularit¨aten und die zugeh¨origen Entwicklungen willk¨urlicher Funktionen, Math. Ann., 68 (1910), 220–269.

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Discrete Versions of Some Classical Integrable Systems and Factorization of Matrix Polynomials1 § 0. Introduction In this paper we study a class of discrete integrable systems which are closely related to problems occurring in mathematical physics such as the Heisenberg model for classical spins or the billiard problem in the interior of an ellipsoid. A discrete system can be viewed as the iterates of a symplectic mapping, the time t ∈ Z being the number of iterations. Such a system will be called integrable if it possesses sufficiently many integrals which are in involution with respect to a symplectic structure. To describe such a discrete system we take as starting point a variational principle δS = 0 for a functional S = S(X) defined on the space of sequences X = (Xk ), k ∈ Z by a formal sum  (Xk , Xk+1 ) . S=

 are points on a manifold  and  is a function on Q k∈Z

 

n 2n = n × n. Here Xk The Euler – Lagrange equation of such a functional are second order difference equations (see Sect. 1) and k ∈ Z plays the role of the discrete time. This description is to be viewed as the discrete analogue of the Hamilton principle δS = 0 for

S=

 (q, q)˙ dt,



and the related symplectic flow can, as usual, be defined via the Legendre transform provided det( q˙q˙ ) = 0. Similarly, one can define a sympleclic structure on Q2n under an appropriate nondegeneracy assumption (see Sect. 1, Sect. 2, (8) and [1]). We will call such a system «integrable» if there are sufficiently many integrals which are in involution with respect to this symplectic structure. 1 Jointly with A. Veselov. Discrete version of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 1991, 217–243.

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As an example of this setup we mention the Heisenberg chain with classical spins, where M = S 2 = {x ∈ R3 , |x| = 1}

 = (x, Jy),

and

where J is a symmetric matrix, which we may take to be diagonal. The corresponding chain of quantum spins 1 (so-called XY Z Heisenberg 2 model) was investigated by Faddeev and Takhtajan [2] in the framework of the quantum inverse scattering method, using the fundamental results by Baxter [3]. As it was shown by Pokrovsky and Khokhlachev [4], the problem of finding some special eigenfunctions in the quantum XY Z-model leads to the stationary equation δS = 0 for the Heisenberg chain with classical spins. For this discrete system Granovsky and Zhedanov [5, 6] found two algebraic integrals and special solutions. The integrability of these systems, even for arbitrary dimension n of the sphere: M = S n , was shown by one of the authors ([7], see also [1]), where the general solution was described in terms of θ-functions, generalizing the connection between the spectral theory of one-dimensional Schr¨odinger operators and the classical Neumann systems derived in [8, 9]. The main problem to be discussed in the first part of this paper is a chain of orthogonal matrices: We take n = O(N ), n = N (N − 1)/2, and



 (X, Y ) = tr(XJY

T

),

where J is a positive symmetric matrix1 . This problem was introduced in [1] where it was shown that in the continuous limit this problem leads to the Euler problem for force-free motion of a rigid body as it was generalized by Arnold lo arbitrary dimensions. Alternately, one could use the positive Lagrangian 1 tr((X − Y )J(X −



2

−Y )T ). Indeed, for X, Y ∈ (N ) this expression agrees with tr J −tr(XJY T ), i. e. differs from − (X, Y ) only by a constant. For the cases N = 3, N = 4 this system was shown to be integrable by explicit construction of commuting integrals [1]. In Sect. 3 we will establish the integrability of this system for all N by using the discrete version of the isospectral technique, which leads to the complete description of the dynamics



1 We denote the «moment of inertia» by J and not, as is customary, by I, to avoid confusion with the identity matrix

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of this system in terms of Abelian functions. The flow is quasi-periodic in the discrete time parameter k and is linear on a Prym variety. We indicate the approach underlying the solution of this problem. It is based on the construction of an isospectral mapping on a class of matrices (L) into itself, analogous to the Lax approach in the continuous case in which the differential equation is cast in the form L˙ = [L, A] for some class of linear operators (L). Finding the class of matrices (L) and the isospectral deformation is a hit-ormiss game and depends on good guesses. In the discrete case it turns out to be connected with a factorization of matrices. We recall the beautiful observation by Symes [11] that the QR-algorithm of Jacobi matrices is closely linked to the Toda flow: The QR-algorithm, an important device in numerical analysis for diagonalization of matrices, consists in factoring a real nondegenerate matrix L into a product L = QR of an orthogonal matrix Q and an upper triangle matrix R with positive diagonal elements. Now the mapping L = QR → L1 = RQ = Q1 R1 gives rise to an isospectral map since L1 = Q−1 LQ. It was Symes’ observation [11] that the application of this process to L = exp K, where K is a symmetric tridiagonal (or Jacobi-) matrix leads to an integrable mapping which is interpolated by the Toda flow. This idea has been extended to more general classes of matrices by Deift et al., see [12]. Although this mapping has no variational description, this idea turns out to be fruitful also for the problems at hand. The main new feature is that we start with a class of certain quadratic matrix polynomials L(λ) = A0 + A1 λ + A2 λ2 and a suitable factorization L(λ) = (B0 + B1 λ)(C0 + C1 λ) = B(λ)C(λ). If such a factorization exists and can be defined in a unique way then it gives rise to an isospectral mapping L(λ) → L1 (λ) by exchanging the factors: L1 (λ) = C(λ)B(λ) = B1 (λ)C1 (λ) = B −1 (λ)L(λ)B(λ). This can be viewed as a discrete analogue of the Lax representation.

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The main difficulty is, of course, to find the class of matrix polynomials together with a factorization in such a way that it corresponds to the dynamics of the given problem. Moreover, even if one has such a factorization it is generally not unique and the above procedure gives rise only to a correspondence, i. e. a multiple valued mapping; this is in good agreement with the fact that frequently the difference equation δS = 0 gives rise to such correspondences. In the problem of orthogonal chains we are able to describe such a class of matrix polynomials and a corresponding factorization which can be made unique by specifying a suitable splitting of the spectrum. From this Lax representation we will find the integrals as well as the algebraic curve on whose Jacobian variety the flow becomes linear in k. A general theory of factorization of matrix polynomials can be found in [13]. The special factorization derived here (see Sect. 1) uses standard ideas from [13]; it involves orthogonal matrices and may be of interest in itself. Using the ideas of «finite-gap» integration [14] in the matrix case developed by Dubrovin [15] we exhibit explicit formulas for the dynamics in the case n = 3 in terms of classical elliptic functions. In Sect. 2 we discuss some generalizations of this system, including the as the above mentioned Heisenberg model with classical spins. We take manifold of rectangular n × N matrices X, n  N for which XX T = In and define the Lagrange function by



 (X, Y ) = tr(XJY

T

),



where J is a symmetric N × N matrix. In other words is the Stiefel manifold Vn, N of orthonormal n-frames in RN . For n = 1, N = 3 this represents the Heisenberg model and for n = N we obtain the chain of orthogonal matrices described above. For n = 1 we show how the factorization procedure leads to the hyperelliptic curves and formulas involving θ-functions as in [1]. In the general case 1 < n < N we exhibit the corresponding factorization problem without full treatment. The last section (Sect. 3) is devoted to the billiard problem in the interior of an ellipsoid in RN . This problem also fits into the above framework. The relevant class of matrices for the Lax representation agree with those introduced in [10] for the study of the geodesic flow on an ellipsoid — probably the oldest nontrivial integrable system in arbitrary dimensions. This class of matrices fits into the procedure of Sect. 1. Finally, we will establish a connection between this billiard problem and a discrete version of the Neumann problem, where a certain symmetry of this system, found in [1], will play a crucial role. In the continuous case such a connection between the geodesic flow on the ellipsoid

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and the Neumann system was discovered by Kn¨orrer with the help of the Gauss mapping of the ellipsoid [16]. However, the connection described here is of a different nature. In the above discussion we referred frequently to a discrete version of a continuous system, as in the case of the billiard problem inside an ellipsoid and the geodesic flow on the same ellipsoid — whose orbits are obtained as limits of tangential billiard shots. Another example is the above mentioned chain of orthogonal matrices and the corresponding continuous system of the force free top. Without trying to be precise we require in all these cases a) that the discrete system tends to the continuous system under a limit process and b) that both systems are integrable and are given by «natural» variational problems. As a rule it is easy to go from a discrete system to a continuous one without destroying integrability. However, the converse is much more difficult, as is often the case if one wants to preserve some structure under discretization of a continuous system. Of course, one could take the «time ε» mapping of a flow but that is usually not described by a simple variational problem. The difficulty is to preserve the integrability under discretization. In this sense our method may be of interest since it provides an approach to the construction of an integrable discretization for the continuous system with known Lax representation, polynomially depending on an additional «spectral» parameter λ. The importance of representations of this type became clear after the paper of Novikov [17]. They exist for most of the known integrable hamiltonian systems and are related to the theory of Lie algebras (see [18, 19]). From this point of view the nature of the factorization procedure calls for a better understanding. Notice that in all our examples the matrix polynomials L(λ) have the property L∗ (λ) = L(λ), where L∗ (λ) = LT (−λ) and the corresponding factorizations L(λ) = B(λ)C(λ) satisfy the condition B(λ) = C ∗ (λ). In a forthcoming paper [29] it is shown that the factorization of certain linear(!) matrix polynomials, introduced in [10], leads to the billiard problems in domains on the sphere and the Lobachevsky space, bounded by conic sections. The present paper was completed in February 1989 and has been circulated as a preprint of the Forschungsinstitut f¨ur Mathematik Z¨urich. For various reasons its publication has been delayed. We added some relevant new references at the end of this revised paper. In particular, we draw attention to the interesting work by Deift, Li, and Tomei [32] in which the systems considered in the present paper are related to loop groups. Moreover, it is shown how the discrete mappings considered here can be interpolated by integrable Hamiltonian flows.

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§ 1. The Discrete Version of the Dynamics of a Rigid Body 1.1. The Equations of «Motion» We consider the functional S(X), determined by a formal sum  T S= tr(Xk JXk+1 )

(1)

k

on the sequences X = (Xk ) with Xk ∈ O(N ), i. e. orthogonal N by N matrices. The stationary points of S are described by the equation δS = 0 or Xk+1 J + Xk−1 J = Λk Xk ,

(2)

where Λk = ΛTk is a matrix Lagrange multiplier, which is determined in such a way that Xk XkT = I. Λk is uniquely determined by Xk−1 , Xk , Xk+1 but as we will see later, not uniquely by Xk−1 , Xk ; it is a complicated function of Xk−1 , Xk . Therefore we use the Euler description of the dynamics. This can be done in the following way. Rewrite (2) as T T . + Xk JXk−1 Xk+1 JXkT + Xk−1 JXkT = Λk = ΛTk = Xk JXk+1

(3)

T Introducing mk = Xk JXk−1 − Xk−1 JXkT we see that the last Eq. (3) means that mk+1 = mk . The conservation of mk , which is the discrete analogue of the angular momentum in space [20] is the consequence of the left-invariance of (X, Y ) (see [1]). In the variables fixed relative to the body we have the «angular velocity» ωk = XkT Xk−1 = Xk−1 Xk−1 ∈ O(N ) and «angular momen−1 mk Xk−1 = ωkT J − Jωk ∈ O(N ) tum with respect to the body» Mk = Xk−1 and thus Eq. (3) can be rewritten as a «discrete Euler – Arnold equation» [1]  Mk+1 = ωk Mk ωk−1 , (4) Mk = ωkT J − Jωk , ωk ∈ O(N ).



In the continuous limit when Xk = X(tk ), tk = t0 + kε, ωk = Xk−1 Xk−1 ≈ ≈ I − εΩ(tk ), ωk = Xk−1 Xk−1 and Mk = ωkT J − Jωk ≈ ε(JΩ + ΩJ) = = εM (tk ), M = JΩ + ΩJ, this Eq. (4) becomes the usual Euler – Arnold equations for the motion of the N -dimensional rigid body  M˙ = [M, Ω], (5) M = JΩ + ΩJ, Ω ∈ o(N ).

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The main new feature of the discrete system (4) is the connection between M and ω: (6) M = ω T J − Jω, ω ∈ O(N ), M T = −M,

which we need to solve to find ω. In fact such ω is not unique (see below) and ωk+1 is not uniquely determined by (4), which therefore leads to a correspondence and not to a mapping. We discuss the symplectic properties of this correspondence (see also [1]). The Eq. (2) is a particular case of the Lagrangian equations δS = 0 for the functional  (Xk , Xk+1 ), Xk ∈ n , = (x, y) (7) S= k∈Z



 

  

(see the Introduction), which in an appropriate coordinate system (x, y) on Q2n = n × n can be written as





∂ (X , X ) + ∂ (X k k+1 k−1 , Xk ) = 0. ∂x ∂y

δS = 0,

(8)

The submanifold Γ2n in Q2n × Q2n , defined by x′ = y,





∂ (x′ , y ′ ) + ∂ (x, y) = 0 ∂x ∂y

determines generally some correspondence between subsets of Q2n . On Q2n one can define a closed 2-form σ by



2 dx ∧ dy σ= ∂ ∂x ∂y



or

  σ = dβ = d ∂ (x, y) dx , ∂x



β = ∂ dx, ∂x

 = ∂∂x dx + ∂∂y dy is a natural decomposition of the 1-form on Q

where d

2n

The submanifold Γ2n is isotropic for the form σ ′ − σ on Q2n × Q2n . Indeed   β′ − β



 

    = ∂ (x′ , y ′ ) dx′  − ∂ (x, y) dx = ∂x ∂x Γ2n Γ2n Γ2n       = − ∂ (x, y) dy  − ∂ (x, y) dx = −d (x, x′ ) . ∂y ∂x Γ2n Γ2n Γ2n





.

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We see that is the generating function of the mapping, determined locally by (8) in the domain of nondegeneracy of σ:  2    det ∂  = 0, ∂x ∂y



and therefore this mapping is symplectic with respect to the symplectic structure σ. In this connection it is useful to introduce the discrete version of the Legendre transformation τ of Q2n into T ∗ . It is defined by



τ : (x, y) → (x, p),

p dx =

 (x, y) dx, x



where p is the fiber coordinate and α = p dx the standard 1-form on T ∗ . Thus the pullback of this form is τ ∗ α = β = x dx, and the standard symplectic form dα on T ∗ pulls back to τ ∗ dα = dβ = σ, which is nondegenerate whenever τ is noncritical. Generally, τ has only a local inverse. We discuss the concepts in our case where = O(N ), (X, Y ) = = tr(XJY T ) and β = tr(dXJY T ).









To describe the Legendre transformation τ : O(N ) × O(N ) → T ∗ O(N ) we identify T ∗ O(N ) and T O(N ) via the bilinear form tr(AB T ) so that τ : (X, Y ) → (X, P ),

P = Y J − XS ∈ TX O(N ),

where S = S T is so chosen that X T P is skew symmetric, i. e. P = 1 (Y J − XJY T X). 2 The standard 1-form α = tr(P T dX) is taken into β = tr(dXJY T ), since X T dX is skew symmetric. The standard sympletic form dα on T ∗ O(N ) is mapped into σ = dβ = tr(dXJ ∧ dY T ). We record that this 2-form a is the pullback of the standard symplectic form on T ∗ O(N ) under τ ; it is nondegenerate at all noncritical points of τ . If we define locally a mapping ϕ : (X, Y ) → (X ′ , Y ′ ) by selecting a branch of the correspondence Y ′ J + XJ = ΛX ′ ,

X ′ = Y,

ΛT = Λ,

(9)

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where X, Y , X ′ , Y ′ ∈ O(N ), then this mapping preserves σ. Equivalently, the mapping τ ϕτ −1 , locally defined near regular values of τ , preserves the standard symplectic structure as well as the corresponding Poisson structure on T ∗ O(N ). Since both τ , ψ commute with left translation of O(N ) we can reduce the map τ ϕτ −1 to a mapping of o∗ (N ) by projecting (X, P )→X T P ∈ o(N ). The resulting reduced mapping ψ : o∗ (N ) → o∗ (N ) is the one defined by (4) taking M = Mk into M ′ = Mk+1 , i. e. ψ : M = ω T J − Jω → M ′ = Jω T − ωJ.

(10)

T

Here it is crucial to solve the matrix equation M = ω J − Jω for ω ∈ O(N ), a question which will be discussed completely in Subsect. 1.2. Now it is well known that the reduction of T ∗ O(N ) to o∗ (N ) takes the standard Poisson structure of T ∗ O(N ) (up to a constant nonzero factor) into the Lie – Poisson structure {f, g} = tr(M [fM , gM ]),

f, g ∈ C ∞ (o(N ))

(11)

on o∗ (N ) which we identify again with o(N ); here fM denotes the skew-symmetric matrix of partial derivatives ∂f /∂Mij . This proves that the mapping ψ : M → M ′ of (4) preserves this Poisson structure, which agrees with the Poisson structure preserved by the usual continuous rigid body motion given by (5). This reduction is the discrete version of the well known reduction procedure [20] for Hamiltonian systems. For the derivation of (11) see also [30, 31]. Our next goal is to show that this mapping is integrable, i. e. preserves sufficiently many functions Fi , which are in involution with respect to the above Poisson structure. As a matter of fact it turns out that these «integrals» have the same form as in the continuous case which are known to be in involution. 1.2. The Solution of the Matrix Eq. (6): ω T J − Jω = M We have to solve two crucial problems: a) to define the mapping ϕ in a unique way by selecting a branch of the correspondence and b) to verify that this mapping is integrable. Both these problems can be reduced to an appropriate factorization problem for a matrix polynomial, as we will show now. The first problem, to construct a well defined map ϕ : (X, Y )→(X ′ , Y ′ ) whose graph belongs to the correspondence (9) reduces to finding a well defined solution ω ∈ O(N ) of the matrix equation: M ′ = (ω ′ )T J − Jω ′

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for a given skew-symmetric matrix M . Indeed, setting ω = Y T X, M = ω T J − − Jω, M ′ = ωM ω −1 = Jω T − ωJ, then X ′ , Y ′ of (9) is given by X ′ = Y , Y ′ = X ′ (ω ′ )T . This problem is, in fact, equivalent to finding a definite inverse for the Legendre transformation τ : (X, Y ) → (X, P ), since X T P = 1 (X T Y J − JY T X) = 1 (ω T J − Jω). 2 2 Hence a solution ω of this equation gives rise to Y = Xω T , thus defining τ −1 . The crucial observation is contained in the following Lemma. The matrix Eq. (6) is equivalent to the factorization (I − λM − λ2 J 2 ) = (ω T + λJ)(ω − λJ).

(12)

The proof is an obvious verification, which shows also that the solution ω is necessarily an orthogonal matrix. It turns out that the choice of the solution ω is fixed by the corresponding factorization of the determinant P (λ) = det(I − λM − λ2 J 2 ) = p(−λ)p(λ).

(13)

We will prove below Theorem 1. Assume that for the real skew symmetric matrix M the polynomial P (λ) = P (−λ) admits a splitting P (λ) = p(λ)p(−λ); with a real polynomial p(λ) satisfying |p(λ)| + |p(−λ)| > 0

for all λ ∈ C

then there exists a unique matrix ω ∈ O(N ) satisfying (12) and ±p(λ) = det(ω − λJ). We postpone the proof of this theorem to Subsect. 1.4. We discuss the splitting of the determined P (λ). Since M + M T = 0 one has P (λ) = P (−λ) and the set Σ of all roots of P (λ) satisfies Σ = −Σ. The factorization (13) corresponds to a splitting Σ = Σ+ ∪ Σ− into disjoint sets Σ+ , Σ− satisfying Σ + = Σ+ ,

Σ − = Σ− ,

Σ+ = −Σ− ,

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where Σ+ is the zero set of the real polynomial p(λ). Here we denote for any set A ∈ C by A the set of all a, a ∈ A and by −A the set of (−a), a ∈ A. Any such splitting gives rise to such a factorization (12) and thus to a solution of (6). Obviously the possibility of such a factorization requires that P (λ) has no roots on the imaginary axis. In this case one factorization is obtained by taking for Σ+ the roots of P (λ) in the right half plane, and Σ− = −Σ+ . We give an outline of the proof of Theorem 1 under the assumption that the roots of P (λ) are distinct, leaving the complete proof of the general case for later. Denote the elements of Σ+ by λ1 , λ2 , . . . , λN (and Σ− = = {−λ1 , −λ2 , . . . , −λN }). Then there exist eigenvectors ψk : (I − λk M − − λ2k J 2 )ψk = 0. Because of the nondegeneracy of ω T + λi J we have from the factorization (12) (ω − λk J)ψk = 0, or, equivalently ωψ = JψΛ, where ψ is the N by N matrix with columns ψk and Λ= diag(λ1 , λ2 , . . . ,λN ). If ψ is invertible then (14) ω = JψΛψ −1 defines the desired solution. It can be shown that ψ is indeed nondegenerate and that (14) actually defines the solution of (6) which, moreover, is real and orthogonal. But in Subsect. 1.4 we present another approach to the solution of (6) which is a bit more general. This proof will also provide the nondegeneracy of ψ and complete the above considerations. In this connection the concepts of symplectic geometry will turn out to be useful. We note that the solutions ω ∈ O(N ) so obtained have the property that the polynomials p(λ) = ± det(ω − λJ) and p(−λ) have no common roots. In other words, any two eigenvalues λ, λ′ of ωJ −1 satisfy λ + λ′ = 0. We will denote the set of these matrices by E, i. e. E = {ω ∈ O(N ), |p(λ)| + |p(−λ)| > 0 ∀λ ∈ C; p(λ) = det(ω − λJ)}.

This is clearly an open subset of O(N ) containing a neighborhood of the identity. Since p(λ) does not vanish on the imaginary axis E decomposes into several components, depending on how many roots of p(λ) lie in the left half plane. With the aid of Theorem 1 it is easy to define a mapping ϕ in a unique way. We do this in the reduced form and rewrite the above factorization (12) in the

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form (I − λM − λ2 J 2 ) = AT (−λ)A(λ);

Then the image point M ′ = ψM is given by

A(λ) = ω − λJ.

(I − λM ′ − λ2 J 2 ) = A(λ)AT (−λ),

(16)

(17)

where the two factors were exchanged. This equation is readily verified from (10). Thus the determinants P (λ), P ′ (λ) of (16), (17) respectively are identical. By Theorem 1 any splitting P (λ) = p(λ)p(−λ) gives rise to a unique factorization. Hence for (17) there exists a unique ω ′ and A′ (λ) = ω ′ − λJ with T

(I − λM ′ − λ2 J 2 ) = A′ (−λ)A′ (λ) with det(ω ′ − λJ) = det(ω − λJ) = p(λ).

(18)

′

This gives rise to a well defined mapping ω → ω taking E into itself. The uniqueness is achieved by the requirement (18) which is consistent with iterations of the mapping1. Similarly, the mapping ϕ : (X, Y ) → (X ′ , Y ′ ) is well defined on the left invariant set  = {X, Y ∈ O(N ), Y T X ∈ E} Q and given by (X ′ , Y ′ ) = (Y, Y (ω ′ )T ) if ω = Y T X. We mention that one easily  is precisely the set of regular points of the Legendre transform τ . verifies that Q  making Q  a symplectic manifold. Thus σ is nondegenerate on Q

1.3. Isospectral Deformations

From the above considerations we obtain the desired integrals. For this purpose we write the mapping in terms of an deformation. The Eq. (4) * ) isospectral N «trivial» integrals tr(M 2ν ), ν = is already in this form but yields only k = 2

= 1, 2, . . . , k, (in fact, these are coadjoint invariants of O(N )) which is not

1 We note that this mapping ω → ω ′ in E is the product of two involutions: We observe that to any ω ∈ E ⊂ O(N ) we can, by Theorem 1, associate a second solution ω∗ of (6) for which spec(ω∗ J −1 ) = − spec(ωJ −1 ), i. e. for which p∗ (λ) = det(ω∗ − λJ) = (−1)n p(−λ). The map j1 : ω → ω∗ is clearly an involution on E. To bring the spectrum back we use the trivial involution j2 : ω → −ω T . One verifies readily that our mapping is given by j1 ◦ j2 , i. e. ω ′ = j1 j2 ω.

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sufficient for complete integrability. As was pointed out by Novikov [17] it is crucial to have such a representation for a matrix depending polynomially on a parameter λ. In our case we make use of (16), (17) to obtain for the mapping ψ : M → M ′ the form (I − λM ′ − λ2 J 2 ) = A(λ)(I − λM − λ2 J 2 )A−1 (λ) or equivalently L′ (λ) = M ′ + λJ 2 = A(λ)(M + λJ 2 )A−1 (λ).

(19)

Consequently the polynomials fk (M, λ) = tr(M + λJ 2 )k are integrals of ψ. In other words, the characteristic polynomial det(L(λ) − µI) is preserved by ψ, or in homogeneous form  ν 2α λβ µγ Qαβγ (M ), det(νM + λJ 2 − µI) = 2α+β+γ=N

the coefficients Qαβγ (M ) for α  1, 2α + β + γ = N provide k 2 integrals if N = 2k, or k(k + 1) integrals if N = 2k + 1. These integrals fk (M, λ) or Qαβγ (M ) are precisely the same as for the Euler – Arnold Eq. (5). Indeed, for these equations the Lax representation was found by Manakov [22] in the form d (M + λJ 2 ) = [M + λJ 2 , Ω + λJ] dt showing that also fk (M, λ), or Qαβγ (M ) are integrals of the motion. It is well known that these functions are in involution with respect to the Poisson structure (11) and independent, making the system (5) completely integrable. Since our discrete map ψ : M → M ′ preserves the same Poisson structure (11) as well as the functions fk (M, λ) we conclude that ψ is also integrable. We summarize these results in Theorem 2. The discrete Euler Eq. (4) is equivalent to the isospectral deformation det Ak+1 = det Ak , Lk+1 = Ak Lk A−1 k , where Lk = Mk + λJ 2 , Ak = ωk − λJ, Mk+1 = ψ(Mk ). This mapping ψ preserves the Poisson structure (11) and is completely integrable. It preserves

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the same Poisson structure and integrals Fi as the continuous system (5) for the motion of the rigid body. The «integration» of this system is now rather straightforward, since the integration%of the continuous case is known: The nonsingular compact level sets Tc = (Fi = ci ) consist of a finite union of tori, according to well known i

arguments [20]. Since our mapping ψ preserves the Poisson structure (11) as well as the functions Fi it commutes with all commuting Hamiltonian flows generated by the Fi , defined by M˙ = [M, ∇Fi ]. On each such torus our mapping ψ must be a translation with respect to the affine structure, determined by these flows. In this case this mapping can be represented as a shift along the trajectory of a certain integral H (see [1] and Subsect. 1.5). We will show that in our case Tc is the real part of a complex Abelian variety of the curve det(M + λJ 2 − µI) = 0, and Eq. (4) determines a translation on it. In fact, it turns out to be the same Prym variety occurring in the integration of the Euler – Arnold equation (see, for example, [21] and [33]). 1.4. The Symplectic Geometry of Eq. (6) First of all we write (6) as ω −1 J − Jω = M,

ωω T = I,

and introducing W = ω −1 J we obtain the quadratic matrix equation W 2 − MW − J2 = 0

(20)

Q(ν) := det(ν 2 I − νM − J 2 ) = 0.

(21)

with the additional condition W T W = J 2 . If ν is an eigenvalue of W then / Σ the splitting Comparing with (13) we see that Q(ν) = ν 2N P (ν −1 ). Since 0 ∈ Σ = Σ+ ∪ Σ− defines the splitting S = S+ ∪ S− of the set S of roots (21): S+ = (Σ+ )−1 ,

S− = (Σ− )−1 .

We require that this splitting satisfies the following conditions: S + = S+ ,

S − = S− ,

S− = −S+ ,

S+ ∩ S− = ∅.

Such splitting exists if (21) has no purely imaginary roots. Notice that now we do allow multiple roots but we do suppose that no root belongs to both components S+ and S− . In particular, purely imaginary roots are excluded. We formulate Theorem 1 in the equivalent form:

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Theorem 1′ . For any splitting S = S+ ∪ S− with the properties (22) there exists a unique solution W of (20) (and therefore the solution of (6) ω = JW −1 ) with spec W = S+ . For the proof the solution of (20) will be played back to a problem of symplectic geometry, namely the determination of invariant subspaces of a linear Hamiltonian vector field. The real 2N × 2N matrix in question is   0 I . A= J2 M We look for an N -dimensional invariant subspace of A:   X z= u, u ∈ RN , Y X, Y being N × N matrices, i. e.     X X A = C Y Y with some real N × N matrix C, with spec C = S+ , or equivalently  Y = XC, J 2 X + M Y = Y C.

If det X = 0,

(23)   x i. e. if the invariant subspace can be viewed as the graph y = Y X −1 x, z = y then we obtain for W = Y X −1 , J 2 + M W = W XCX −1 = W 2 , Eq. (20). To prove that W T W = J 2 , we note that A is antisymmetric with respect to the symplectic bilinear form   M −I [z, w] = (Bz, w), B = (24) I O

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and Az can be viewed as the Hamiltonian vector field with Hamiltonian  2  −J O 1 1 2 2 = (Hz, z) = (−|Jx| + |y| ), H = , 0 I 2 2

(25)

since BA = H and z˙ = B −1 Since νI − A =

z

= B −1 Hz = Az.



νI O −J 2 ν −1 I



I −ν −1 I 2 O ν J − νM − J 2



we have det(νI − A) = Q(ν) = det(ν 2 I − νM − J 2 ), and the spectrum of A is S+ ∪ S− . Denote the N -dimensional eigenspaces of A with respect to S+ , S− by V+ , V− , respectively. Because of S + = S+ , S − = S− they are real and since µi + µj = 0 for µi , µj ∈ S+ they are Lagrangian, isotropic spaces with respectively, as respect to the symplectic form [, ] and the symmetric form follows from the following lemma. Lemma. If Eµk = Ker(A − µI)k and µ + ν = 0 then [Eµk , Eνl ] = 0 for all k, l  0. PROOF. By induction on k + l. For k + l = 0 it is trivial and we assume the lemma for smaller values of k + l. Consider ϕ ∈ Eµk , ψ ∈ Eνl and set

so that

ϕ ! = (A − µI)ϕ ∈ Eµk−1 ,

therefore

µϕ = Aϕ − ϕ, !

ψ! = (A − µI)ψ ∈ Eνl−1 , ! νψ = Aψ − ψ,

! = −[ϕ, ! = 0, (µ + ν)[ϕ, ψ] = [Aϕ − ϕ, ! ψ] + [ϕ, Aψ − ψ] ! ψ] − [ϕ, ψ]

hence [ϕ, ψ] = 0.

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Corollary. The subspaces V+ , V− are Lagrangian: [V+ , V+ ] = [V− , V− ] = 0 and isotropic with respect to

(26)

(z) = 1 (Hz, z) = 0 for z ∈ V± .

:

2

The first statement follows immediately from the lemma because V+ =

span Eµs ,

s=1, ..., N

µs ∈ S+ ,

Eµ = Ker(A − µI)2N .

To prove that V± is isotropic with respect to

consider for ϕ ∈ Eµ , ψ ∈ Eν ,

(Hϕ, ψ) = [Aϕ, ψ] = µ[ϕ, ψ] + [ϕ, ! ψ] = 0

where ϕ ! = (A − µI)ϕ ∈ Eµ . Note that V+ , V− are real subspaces since S + = S+ , S − = S− , while Ek are generally complex. Now we return to the proof of Theorem 2. Let z1 , . . . , zn be any basis in V+ ; combining these as column vectors of an N × 2N matrix   X+ Z+ = (z1 , . . . , zn ) = Y+ of rank N we have from AV+ ⊂ V+ AZ+ = Z+ C+ for some real N × N matrix C+ . To prove (23) we use the relations (26), yielding for any u, v ∈ RN 0 = [Z+ u, Z+ v] = (BZ+ u, Z+ v) = (M X+ u − Y+ u, X+ v) + (X+ u, Y+ v). Assuming v is so chosen that X+ v = 0, we set u = C+ v, so that X+ u = = X+ Cv = Y+ v and we find from the above identity 0 = |Y+ v|2 , i. e. Y+ v = 0, hence Z+ v = 0, i. e. v = 0. Therefore det X+ = 0, proving (23). Thus V+ is given by y = W+ x. Since V+ lies on the zero energy surface it follows |Jx|2 − |W+ x|2 = 0

for all x ∈ Rn proving W+T W+ = J 2 , hence ω = JW+−1 is orthogonal. Moreover spec W+ = S+ proving Theorem 1′ .

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1.5. The Integration of the Discrete Euler Equation Now we apply our results to finding the solution of (4), following the procedure which was described in the continuous case by Dubrovin [9, 10]. For the initial data X0 , X1 ∈ O(N ) we define ω1 = X1T X0 = = X1−1 X0 ∈ O(N ) and M1 = ω1T J − Jω1 . As follows from the previous considerations Eqs. (4) define only a correspondence, but if we fix the splitting S = S+ ∪ S− of the roots of the polynomial Q(ν)(ν 2 I − µMk − J 2 ), which in fact does not depend on k, then we have a well defined mapping fS+ , S− : (ωk , Mk ) → (ωk+1 , Mk+1 ). In order to «integrate» this dynamics consider the spectral curve Γ: det(M + λJ 2 − µI) = 0,

M = M1 .

(27)

We will assume that J 2 has distinct eigenvalues different from zero: Ji2 = Jj2 for i = j, and Ji2 = 0. For generic M , Γ has a genus g =

(N − 1)(N − 2) . The 2

eigenvector ψ(λ, µ),

(M + λJ 2 − µI)ψ(λ, µ) = 0

(28)

normalized by the condition ψ1 + · · · + ψN = 1

(29)

is meromorphic on Γ whose poles define a divisor  = 1 + · · · + g+N −1 (see [10, 16]). In the points at infinity Pi ∈ Γ, where µ ≈ λJi2 , λ → ∞, (i = 1, . . . , N ) we have (30) ψ i (Pj ) = δji . This means that ψ i (λ, µ) is the basis of the linear space of meromorphic functions with the pole divisor  , determined by the conditions (30). In our case M is skew symmetric, therefore Γ has a symmetry σ : Γ→Γ, σ 2 = id: σ(λ, µ) = (−λ, −µ). (31) The divisor  also is not arbitrary because of the following proposition. Denote ψ T (−λ, −µ) by ψ ∗ (λ, µ) and fix λ ∈ C such that the eigenvalues µ1 , . . . , µN of M + λJ 2 determined by (27) are distinct.

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Proposition. Let µ′ = µ be two distinct eigenvalues of M + λJ 2 , then ψ ∗ (λ, µ′ )ψ(λ, µ) = 0.

(32)

For µ′ = µ this product is different from zero: ψ ∗ (λ, µ)ψ(λ, µ) = 0.

(33)

To prove this consider the product ψ ∗ (λ, µ′ )(M + λJ 2 )ψ(λ, µ) = µψ ∗ (λ, µ′ )ψ(λ, µ). On the other hand ψ ∗ (λ, µ′ )(M + λJ 2 )ψ(λ, µ) = = −((M − λJ 2 )ψ(−λ, −µ′ ))T ψ(λ, µ) = µ′ ψ ∗ (λ, µ′ )ψ(λ, µ). We see that if µ′ = µ then ψ ∗ (λ, µ′ )ψ(λ, µ) = 0. But ψ(λ, µ) for all possible µ = µ1 , . . . , µN form a basis, therefore ψ ∗ (λ, µ′ )ψ(λ, µ) can’t be equal to zero. Corollary. The divisor  of the poles of ψ satisfies the equation

 + σ() ≈ B,

(34)

where B is the set of branch points of µ as a function of λ, and ≈ means linear equivalence of divisors. This equivalence is given by the function F (λ, µ) = ψ ∗ (λ, µ)ψ(λ, µ) as follows from the proposition. Thus  belongs to the shifted Prym variety P ⊂ J(Γ). We restrict ourselves to these considerations because the detailed discussion of the algebraic-geometric aspects of this spectral problem can be found in the literature (see [21] and references therein). The corresponding problems of real algebraic geometry are considered in [23]. Now we use the representation (19) for describing the analytic properties on Γ of ψk for arbitrary k. Fix some splitting Σ = Σ+ ∪ Σ− . As follows from (19)  ψ(λ, µ) = (ωk − λJ)ψk (λ, µ)

(35)

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is an eigenvalue of Mk+1 + λJ 2 :   (Mk+1 + λJ 2 )ψ(λ, µ) = µψ(λ, µ).

This means that we can define ψk+1 as

ψk+1 = (ωk − λJ)ψk .

(36)

Notice that ψk+1 does not satisfy Dubrovin’s normalization (29) which is required only for ψ1 . One can see from (36) that ψk+1 has N new poles at the «infinities» P1 , . . . , PN . In order to find the new zeros consider the hyperbola , determined by the equation λµ = 1 and the intersection µ = λ−1 and

∩ Γ. This intersection is described by the equations

det(M + λJ 2 − λ−1 I) = (−λ)−N det(I − λM − λ2 J 2 ) = 0 which coincides with (11). + So we have 2N points of intersection which we denote Q+ 1 , . . . , QN , − − Q1 , . . . , QN in agreement with the splitting Σ = Σ+ ∪ Σ− . As follows from the construction of ωk (see Sect. 3) (ωk − λi I)ψk (Q+ i ) = 0. + This means that Q+ 1 , . . . , QN are the new zeros of ψk+1 . Thus we have proven

Lemma. For a given splitting Σ = Σ+ ∪ Σ− the vector eigenfunction ψk+1 (36) of the matrix Mk+1 + λJ 2 has on spectral curve Γ the following analytical properties, which determine ψk+1 uniquely: 1. ψk+1 has a simple pole in depending on the initial data M1 and the poles at the «infinities» P1 , . . . , PN with asymptotics in



i Pj : ψk+1 = λk (−Jj )k (δji + O(λ−1 )), + 2. ψk+1 has a zero of order k in Q+ 1 , . . . , QN .



λ → ∞.



In particular, we see that the pole-divisor k+1 of ψk+1 is connected with k by the relation (38) k+1 ≈ k + U, where U = P1 + · · · + PN −





Q+ 1

− · · · − Q+ N.

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For given ψk+1 one can reconstruct ψk+1 by using the formula (14), and T Mk+1 as ωk+1 J − Jωk+1 . To find the solution of (2) Xk ∈ O(N ): Xk−1 = ωk ωk−1 . . . ω1 X0−1 , one can use again Eq. (36). Indeed, from (36) follows that Φk+1 = ωk Φk ,

(39)

where Φk is N × N matrix with ψk (0, µi ) as a column. This means that Φk+1 = ωk ωk−1 . . . ω1 Φ1

−1 and Xk−1 = Φk+1 Φ−1 1 X0 .

(40)

For ψk+1 one can write the explicit formulas in terms of Prym’s θ-functions as it was done, for example, by Bobenko in [24]. But here we restrict ourselves to the example of O(3) (see below). We summarize the result of this section in the following theorem. Theorem 3. The discrete Euler Eq. (4) corresponds to the shifts on the + Prym variety P ⊂ J(Γ) (34) by the vector U = P1 +· · ·+PN −Q+ 1 − · · · − QN , depending on the splitting Σ = Σ+ ∪ Σ− . If such splitting is fixed the general solution of (4) and (2) can be expressed as some abelian function on P in the points zk = z0 + kU . 1.6. Explicit Formulas for the Discrete Dynamics of the 3-Dimensional Rigid Body We consider here Eqs. (2), (4) for N = 3. In this case the solution can be expressed by elliptic functions. The spectral curve Γ (27) has the equation  2  λJ1 − µ M12 M13   M23  = 0 det  −M12 λJ22 − µ  −M13 −M23 λJ32 − µ

or

(λJ12 − µ)(λJ22 − µ)(λJ32 − µ) + Hλ − M 2 µ = 0,

2 J32 M12

2 J22 M13

where H = + In the new variables

+

2 J12 M23 .

x = µ/λ,

y = λ,

(41) has the form y 2 Q(x) = H − M 2 x

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C o p y rig h t ОАО « ЦКБ « БИБКОМ» & ООО « A ге н тствоK н ига- C е рвис»

266

Discrete Versions of Some Classical Integrable Systems

with Q(x) = (x − J12 )(x − J22 )(x − J32 ). After another change of variables w = Q(x)y = Q(µ/λ)λ we obtain the standard form of the elliptic curve w2 = R(x) = (x − J12 )(x − J22 )(x − J32 )(H − M 2 x).

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The involution σ : (λ, µ) → (−λ, −µ) in these variables is σ(w, x) = (−w, x) and the Prym variety coincides with J(Γ) ≈ Γ. The «infinities» P1 , P2 , P3 correspond to the branch points x = J12 , x = J22 , x = J32 . The fourth branch point x = H/M 2 corresponds to the point λ = µ = 0, so we choose it as the zero 0 on Γ. Let x1 < x2 < x3 < x4 be the ordered roots of R(x), i. e. the numbers J12 , J22 , J32 , H/M 2 (notice that min{Ji }  H/M 2  max{Ji }). The elliptic integral (x, w) dx z= # R(x) 2 H/M

gives the equivalence of Γ and C/Zτ1 + Zτ2 , where τ1 = 2

=2

x3

x2

x2

x1

dx ; τ is real, τ is purely imaginary. # 1 2 R(x)

dx , τ = # 2 R(x)

The equation µλ = 1 in the variables x, w has the form (x − J12 )(x − J22 )(x − J32 ) − x(H − M 2 x) = 0;

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+ + − it determines on Γ the set of six points, which we denote as Q+ 1 , Q2 , Q3 , Q1 , − − Q2 , Q3 according to the splitting Σ = Σ+ ∪ Σ− (see Fig. 1). This figure depicts the situation, corresponding to J12