Fortschritte der Physik / Progress of Physics: Band 26, Heft 1 1978 [Reprint 2021 ed.] 9783112559307, 9783112559291

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

26. BAND 1978

A K A D E M I E

- V E R L A G

•

B E R L I N

Inhalt des 26. Bandes Heft 1 Variational Matrix Padé Approximants in Two Body Scattering J., Modification of Dispersion Relations and Generalization of the Hartree-Fock Method for Hard-Core Interactions in Nuclear Matter

TURCHETTI, G.,

1

WINTER,

29

Heft 2 SALAM,

A., and J .

STRATHDEE,

Supersymmetry and Superfields

57

Heft 3 D E ALFARO, V . ,

and M.

RASETTI,

BOYA, L . J . , J . F . CARINENA,

and

Structural Stability Theory and Phase Transitions Models J . M A T E O S , Homotopy and Solitons

143 175

Heft 4 J . J., and M. M. N A G E L S , The Nucleon-Nucleon Scattering 215 T., und W. B R A U N E , Deformationstheorie des Ladungsträgerspektrums im Wismut 2 4 1 L O P U S Z A N S K I , J . T., The Representations of the Poincaré Group in the Framework of Free Quantum Fields 261 D E SWART, JUNG,

Heft 5 DITTRICH,

W., Source Methods in Quantum Field Theory

. 289

Heft 6 W A D ATI, M . , M . M A T S U M O T O , Y . T A K A H A S H I ,

and

H . UMEZAWA,

Crystals and Dislocations

Quantum Field Theory of 357

Heft 7/8 L., Linear Fermion Systems, Molecular Field Models, and the KMS Condition 3 9 7 B., Quantum Field Theory with Soliton Conservation Laws . 441

VAN HEMMEN, SCHROER,

Heft 9 FROLOV, V. P., Null Surface Quantization and Quantum Field Theory in Asymptotically Flat Space-Time 455 DAFTARI, I . K . , D . K . BHATTACHARJEE, S. C . N A H A , D . C . GHOSH, and T . R O Y , On the Systematica of the Number of Heavy Prongs in Proton-Nucleus Interactions in Emulsion. . . 501

Heft 19 LANG, C. B., The tcK Scattering and Related Processes

509

Heft 11/12 KLEINERT, H . ,

Collective Quantum Fields

565

ISSN 0015-8208

FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN D E M O K R A T I S C H E N R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. R I T S C H L UND R. ROMPE

H E F T I - 19 7« - B A N D 26

A K A D E M T E

- V E R L A G

EVP 1 0 , - M 31728

•

B E R L T N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in ¡der D D R an eine B u c h h a n d l u n g oder an den Akademie-Verlag, D D R - 108 Berlin, Leipziger S t r a ß e 3 - 4 — im sozialistischen Ausland an eine B u c h h a n d l u n g f ü r fremdsprachige L i t e r a t u r oder an den zuständigen Postzeitungsvertrieb — in der B R D und Westberlin an eine B u c h h a n d l u n g oder an die Auslieferungsstelle K U N S T U N D W I S S E N , Erich Bieber, 7 S t u t t g a r t 1, Wilhelmstraße 4—6 — in Osterreich a n den Globus-Buchvcrtrieb, 1201 W i e n , H ö c h s t ä d t p l a t z 3 — im übrigen Ausland an den I n t e r n a t i o n a l e n Buch- u n d Zeitschriftenliandel; den B u c h e x p o r t , Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 701 Leipzig, P o s t f a c h 160, oder a n den Akademie-Verlag, D D R - 108 Berlin, Leipziger S t r a ß e 3 - 4

Zeitschrift „Fortschritte der Physik* 1 Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Ldschc, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 223 62 21 und 2236229; Telex-Nr. 114420; Postscheckk o n t o : Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Aaschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR » 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik'* erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je Heft 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses H e f t e s : 1027/26/1. (c) 1978 by Akademie-Verlag .Berlin. Printcd in the German Dcinocratic Republic. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 26, 1 - 2 8 (1978)

Variational Matrix Pade Approximants in Two Body Scattering G . TUBCHETTI*

Centre for Interdisciplinary Studies in Chemical Physics, The University of Western Ontario, London, Canada Abstract The convergence and bounding properties of the variational matrix Pade approximants are investigated for non relativistic two body interactions. Selecting L — 1 discrete values qi, i = 1 , . . . , L — 1 and the physical momentum q0 the off shell scattering amplitudes are Lx L matrices. The [N/N] Pade approximants to the Born series of these matrices are the variational solution of the Schwinger principle and the corresponding physical amplitude has variational properties in the off shell momenta. For positive interactions the best approximants to the phase shift is an absolute minimum on the qi and monotonic convergence to the exact result for N - > co or I —> oo ca be proved. Similar properties are shown for the bound states using the Ritz variational principle. The required mathematical background is extensively worked out, the extensions to non positive, singular and long range potentials are considered and some numerical examples are presented.

Introduction The two body interaction in potential scattering has been constantly investigated, even though the problem might seem of academic nature, since it provides a good test for new approximation methods. The basic program is to obtain the phase shifts and bound states starting from the Born series of a scattering amplitude, through an approximation scheme which can be extended, at least formally, to the many body problem, to relativistic interactions and to quantum field theory. The Pade approximants allow such extensions and enjoy the correct unitarity and analyticity properties. Rigourous convergence statements have been proved in potential scattering and for the Bethe Salpeter equation in the ladder approximation. In field theory, beyond the formal level, one faces formidable difficulties since the Green's functions do not fulfil a finite set of integral equations as they do for any system with a finite number of degrees of freedom. As a consequence starting from a finite expansion we can, in potential scattering, reproduce the exact solution, while in field theory even the exact sum of all the related geometric series will always produce an approximate result. The key words in our scheme are the matrix Pade approximants (P.A.) to the Green's function and the extremal properties in the off shell momenta. We show that for any * Permanent Address: Instituto di Fisica, Universita di Bologna and I N F N sezione di Bologna. 1

Zeitschrift „Fortschritte der Physik", Heft 1

2

G. Ttjbchetti

regular positive potential, the matrix Pade approximants converge providing strict bounds to the phase shift and binding energies and that the results can be improved at any order by using the off momenta as variational parameters. For singular positive potentials a regularization procedure still allows obtaining bounds and monotonic convergence. The same results apply to non positive and long range interactions if the distorted wave technique is properly used. The plan of the work is the following: In section I. after a historical survey we quote the basic equations of the formal scattering theory and examine the analytic properties of the T and K matrices. In section II. we develop the mathematical theory of matrix P.A. starting from the variational principles. The connection with the method of moments is established and the relevant convergence theorems are quoted. In section III. we apply these results to the total and partial wave T and K matrices and discuss the convergence and bounding properties. Section I.

1.1. Historical Survey The Pade approximants in potential scattering were first considered for the scattering length [1], Later it was shown that the P.A. to the partial wave K matrix are the exact solution of separable potentials [2]. The Stieltjes nature of the forward T matrix at negative energies [3] and of the forward K matrix for positive energies was recognized [4] and the denominators or P.A. to the partial wave amplitudes were proved to converge to the Jost functions [-5]. A similar analysis was carried out for the ground state of a system of fermions [6] and for the scalar Bethe Salpeter equation in the ladder approximation [7, S]. The connection with the Schwinger variational principle, using the Cini Fubini ansatz [9], was established [10, 11]. The numerical analysis of the exponential [12] and the Yukawa [13] potential confirmed the efficiency of the method and exhibited some difficulties in the computation of bound states. Difficulties of the same nature had been first noticed in a field theoretical calculation of the deuteron [14], A remarkable progress was made on realizing that the off shell momenta could be used as variational parameters. First pointed out for field theoretical purposes [15] the variational aspect was systematically investigated in potential scattering [16] and the extensions to the three body problem [IT1] and the scalar Bethe Salpeter equation was considered [1.

Z

(i.4)

The transition operator T(z), accounting for (1.2), reads T(z) = (1 - gVG0(z))~i gV = gV(l -

gG0{z)V=

gV + gW(z - H)^ V

(1.5)

nd fulfills t he linear equation T(z) = gV + gVO0(z) T(z) = gV + T(z) G0(z) gV.

(1.6)

We notice that T(z) has an inverse T~\z) for Im 2 =j= 0 and that the reflection property of G0(z), namely G0+(z) = G0(z*) implies T+(z) = T(z*).

(1.7)

It is therefore convenient to introduce an operator K(z) hermitian and continuous on the real positive axis, by " T-\z) + T~\z*) " K{z) = (1.8) 2 The equations fulfilled by K(z) are immediately derived from (1.8) and (1.5) K(z) = gV + gVG0p(z) K(z) = gV + K(z) Gor{z) gV G0p(z) = j

[G0(z) + G0(z*)].

(1.9) (1.10)

Using equations (1.5) and (1.7) we find that the antihermitian part of 2 1 - 1 depends only on H 0 T~\z) — T~1+(z) = 2mP{z) (1.11) where m 1*

= ^iG0(z*)-G0(z)].

(1.12)

4

G. T u r c h e t t i

Since the hermitian part of T _1 (z) is the K ' 1 operator .(1.8) we may finally write T { z )

=

[ K - \ z )

+

wrP(z)]-1.

(1.13)

Equation (1.11) leads for physical energies z = E + iO to the extended unitarity condition. In fact P(E) is spectral density of H0 at the point E and can be formally written = (2*)-»

P ( E )

f

i

t

s

|£) 0 in the angular momentum basis. Labelling by 11, q) a state of angular momentum I and momentum q {H \q, T) = g2/2m|g, I)), the partial wave expansion of the T and K amplitudes reads 0

T ( q ' ,

q,

z)

=

4 »

£

¿=o

(21

+

1)

P ,

\ 11 !

T

t

{q,

ff',

«)

(1-15)

where q = \q\, q' = \q'\ and Ti(q', q, z) = (q', Z[ T(z) \q, I ) . In Appendix A we quote our normalization conditions and give explicitly the orthogonality and closure relations in the angular momentum basis. Using the closure relation on (1.13) we find for the physical energies z = E + ¿0 P {

E )

=

^

L

j r

™

\p,

l)

(jp,l\,

p

=

(1.16)

(2mE)V*

1=0

and the extended unitarity equation (1.16) becomes for z = E + iO T

t

* { q ' ,

q,

E )

-

T

t

( q ' , q,

E )

=

2i(2mp)

T

From (1.17) we see that the physical amplitude terms of a phase shift d[(E) according to

t

* ( q ' , p,

E )

T i ( p , p , E )

T

t

( p , q,

E).

can be parametrized in

TAp, p, E) = - — i - e ^ ' £ > s i n ( 5 ; ( £ ) . 2

(1.17)

(1.18)

mp

For the physical K matrix we get from (1.13) K

t

( p , p,

E )

=

T

t

( p , p,

E )

[1

^

i 2 m p T

l

( p , p,

E)]~i

=

—

J

—

tan

d,(E).

(1.19)

1.4. Analytic Properties in z and g The analytic properties in the energy z \3T\ and coupling constant g [32] are the starting point for any approximation scheme. The analyticity in z is related to the spectral properties of the total hamiltonian H by (1.5): for a real potential V and g real the off shell amplitude T(q', q, z) is meromorphic in the complex z plane cut along the real positive axis. The poles lie on the real negative axis and correspond to the bound states; the second sheet analytic continuation of T can have additional singularities and the poles with Im z < 0, Re z > 0 are associated to the resonances.

Variational Matrix Pade Approximants

5

The off shell amplitude K(q', q, z) is meromorphic with a sequence of poles on the real axis corresponding to bound states and resonances. The analytic properties in g depend on the behaviour of the potential. F o r regular potentials: T(q', q, z) and K(q',

q, z) are meromorphic functions of g.

T(q, q, z) = K(q, q, z) is a Hamburger function of g for F real and 2 < 0. F o r positive interactions F > 0: K(q, q, z) is a Hamburger function of g T(q, q, z) = K(q, q, z) is a Stieltjies function of g for 2 < 0. Choosing L values of the momenta q a , . . . , q L ~i, if we define two matrices ST and J f by Cfa? = K{qa,

« , 0 = 0, 1,

qf,z),

1,

(1.20)

all the previous statements hold for the matrices a n d J f . (See appendix C for more details on Stieltjes matrices). The proof of the above statements is very simple if one writes the scattering amplitudes as the resolvents of completely continuous operators [33]. F o r future reference we quote them T(q',

K(q',

?> 2) = 9(

(1-22)

Ap(z) = — Vll2G0p(z)

(1.23)

where A(z) — — F x ' 2 (? 0 (z) F 1 ' 2 , where

F1/2,

|(r, q, 2)1 ^ IM?)|| \\1 \

S^

(

: ' : i • (226) (J-N + g/*N+1 • • • + g/*2N J VW The proof will be omitted since it is essentially the same as for Theorem I I I . We wish to point out that the result just obtained is equivalent to the resolvent of PLNAPLN and to the [N — 1/iV] matrix Pade approximant. T h e o r e m V. The stationary values of for ip 6 'SLN and ip' £ are matrix elements of the resolvent of ALN = PLNAPLN. Proof We consider the linear equation IW L N ) = Wi> - gAiN\w L N )

(2.27)

and W*! VfLH) = (Va\ (1 + g^LN)' 1 1n) •

( 2 -28)

An explicit solution of (2.27) is obtained if we decompose explicitly \ippLN) on the basis set of S L N \nLN)=ZlN£cnyfA-\tpy). n= 0 y=0

(2.29)

Replacing (2.29) in (2.27) and left multiplying by A+m\cpb) we get E1 E n=0 y= 0

*n+ro

1™,

are the (2.33)

-l/gr^Wr'™)

\ip') = |ipr'LN) L*) = - 1 ¡gr™.

(2.34)

Vr

Proof W e choose \i/i) and |%p') according to lv> =

E 1 z ' K A ^ ) , n=0 a=0

W) = " z m—0 p=0

EK^\n)

L

(2.35)

and notice t h a t when ,

A+N W) = — 1 | V ' >

are satisfied for g = grLN and \ip) = \ip,LN), \ip') = tion (2.35) in (2.40) leads to equations (2.37) q.e.d.

(2.40)

\ipr'LN)- I n fact using the decomposi-

T h e o r e m V I I . T h e stationary values R L N , S L N of the Schwinger functionals are the [N — 1/N] and [AT/JV] matrix Pade approximants (P.A.) to the matrix R^(g).2) RLN = [ N 2

m

w

S ™ = [NIN]% W

) We shall label the matrix P. A. by a superscript corresponding to the order of the matrix.

(2.41)

Variational Matrix Padé Approximants

11

Proof

The theorem is evident for N = 1 and any L. For any L and N the proof is ratherinvolved and will be reported elsewhere [35]. We shall only sketch the proof of L = 1 and any N since the commuting fin make it much simpler. Theorem V implies t h a t R™(g)

=

=

(