Quantum Inversion Theory and Applications 3662139715, 9783662139714

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Lecture Notes in Physics Editorial Board H.Araki Research Institute for Mathematical Sciences Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan E. Brezin Ecole Normale Superieure, Departement de Physique 24, rue Lhomond, F-75231 Paris Cedex 05, France

J. Ehlers Max-Planck-Institut fiir Physik und Astrophysik, Institut fiir Astrophysik Karl-Schwarzschild-Strasse 1, D-85748 Garching, FRG U. Frisch Observatoire de Nice B. P. 229, F-06304 Nice Cedex 4, France K. Hepp Institut fiir Theoretische Physik, ETH Honggerberg, CH-8093 Ziirich, Switzerland R. L. Jaffe Massachusetts Institute of Technology, Department of Physics Center for Theoretical Physics Cambridge, MA 02139, USA R. Kippenhahn Rautenbreite 2, D-37077 Gottingen, FRG H. A. Weidenmiiller Max-Planck-Institut fiir Kemphysik Saupfercheckweg 1, D-69117 Heidelberg, FRG J. Wess Lehrstuhl fiir Theoretische Physik Theresienstrasse 37, D-80333 Mfinchen, FRG

J. Zittartz Institut fiir Theoretische Physik, Universităt Koln Ziilpicher Strasse 77, D-50937 Koln, FRG

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H. V. von Geramb (Ed.)

Quantum lnversion Theory and Applications Proceedings ofthe 109th W.E. Heraeus Seminar Held at Bad Honnef, Germany May 17-19, 1993

Springer-Verlag Berlin Heidelberg GmbH

Editor H. V. von Geramb Theoretische Kemphysik, Universitlit Hamburg Luruper Chaussee 149, D-22761 Hamburg, Germany

ISBN 978-3-662-13971-4

ISBN 978-3-662-13969-1 (eBook)

DOI 10.1007/978-3-662-13969-1 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1994 Originally published by Springer-Verlag Berlin Heide1berg New York in 1994 Softcover reprint of the hardcover 1st edition 1994 Typesetting: Camera ready by author/editor using the TEXfLATEX macro package from Springer-Verlag 58/3140-54321 O- Printed on acid-free paper

Preface In recent decades, the quantum theory of atomic, nuclear and partide physics has made great progress and this is true for the fundamental as well as for the applied branches of these sciences. Presently, QCD is considered to be the best available fundamental theory of strong interactions and their associated field theories. But the fundamental approach is not helpful for few- and many-body nuclear systems. QCD-specific substructures of nucleons remain hidden and, because .of this, a fully microscopic treatment is not efficient. A similar situation exists for macroscopic matter, where the atomic and molecular substructure is only rarely manifest. Therefore, the design of models is of utmost practica! importance in any branch of physics where one wants to describe quantitatively a restricted number of phenomena. The model approach is very economica!. However, a demand for ever-increasing accuracy requires an ever-increasing number of model parameters, and this implies the loss of elegance and of intuitive physical understanding. It was the privilege of mathematics, which puts observables first and determines the cause of them by inversion, to have inverted the causality chain. Our central interest is inversion of the Schrodinger equation for the determination of potentials. It is beyond any doubt that potentials are very efficient quantum mechanical operators and that the underlying physics is well understood. At this point inversion begins replacing the method of parameter fitting of models. Inversion thus has the well-defined task of linking fundamental laws with the requisites of their application in modern technology. This volume covers aspects of Schrodinger equation inversion with the purpose of determining interaction potentials in partide, nuclear and atomic physics. It includes reviews and reports on latest developments in topical mathematics, supersymmetric quantum mechanics, inversion for fixed-f nucleon-nucleon potentials, inversion of fixed-E optical potentials and their generalizations. Also included are some topics from nonlinear differential equations relating to the Schrodinger or other equations of partide, nuclear, atomic and molecular physics which can be solved by inverse scattering transformations. The material collected in this volume is representative of the research status in this very rapidly growing field. Considering that about forty years have passed since the first important steps were taken in the field of quantum inversion by Gelfand, Levitan, Bargmann, Jost, Kohn, Marchenko, Krein, Moses and Newton, just to mention a few, and that applications are only now beginning to receive comprehensive attention, we expect students and young scientists to develop intensified enthusiasm for the very promising field of inversion in general. We hope that these proceedings will foster this development. The conference was organized and generously supported by the Dr. Wilhem Heinrich Heraeus and Elsa Heraeus foundation as the 109th Heraeus Seminar and took place at the Physikzentrum in Bad Honnef, May 17-19, 1993. For the scientific advice in preparing the conference 1 want to thank Professors Rainer Lipperheide and Helmut Leeb. Hamburg, September 1993

H.V. von Geramb

Contents

D.N. Ghosh Roy An overview of some inverse problems and techniques ................... 1 B.M. Levitan A new approach to the inverse Sturm-Liouville problem ............... 14 M.J. Ablowitz, J. Villorroel Remarks on the inverse scattering transform associated with Toda equations .................... .................... ................ 30 T. Aktosun, C. van der Mee Inverse scattering in one dimension for a generalized Schrodinger equation .................... .................... .......... 37 V.E. Troitsky The potentialless approach to the inverse scattering problem ........... 50 A.A. Suzko Multidimensional and three-body inverse scattering problems in the adiabatic representation .................... .................... . 67 G. Levai Solvable potentiala derived from supersymmetric quantum mechanics .. 107 D. Baye Phase-equivalent potentiala from supersymmetric quantum mechanics .127 Q.K.K. Liu Investigation of the partner-potentials from supersymmetric quantum mechanics by bremsstrahlung .................... ........... 139 R. Lipperheide Quantum inversion on the line for neutron specular reflection ......... 149 P.C. Sabatier Exotica of the Schrodinger problem on the line .................... .... 162 H. Fiedeldey Inversion at fixed-energy for nonlocal and algebraic potentiala and N-body spectral inversion .................... .................... 176 K. Amos Heavy ion scattering; A fixed energy inverse problem ................. 200 A. Zielke, W. Scheid Potentiala in the algebraic scattering theory .................... ...... 227 H. Leeb Darboux-Crum-Kre in transformations and generalized Bargmann potentiala .................... ................. 241 B. Apagyi, P. Levay Electron-atom scattering potentiala obtained by inversion at fixed energy .................... .................... ............... 252

VIII

R.S. Mackintosh, S.G. Cooper Nucleon-nucleus potentiala by IP inversion ........................... 266 H.V. von Geramb, H. Kohlhoff Nucleon-nucleon potentiala from phase shifts and inversion ........... 285 H. Kohlhoff, H.V. von Geramb Coupled channels Marchenko inversion for nucleon-nucleon potentiala . 314 M. Jetter, H. Kohlhoff, H.V. von Geramb Off-shell studies of inversion potentiala with bremsstrahlung .......... 342 M. Coz Factorization of the S-matrix into Jost matrices subject to the conditiona of the np-scattering problem ........................... 353 0.1. Kisaev, A.F. Krutov, 0.1. Muravyev, V.E. Troitsky A simple method of factorization of S-matrices into Jost matrices ..... 363 J. Wambach From skyrmions to the nucleon-nucleon potential ..................... 365 A.A. Stahlhofen Positons as singular Wigner-von Neumann potentiala for Schrodinger and Dirac equations ...................................... 382 M. Bla.Zek Multifractals in high energy nuclear collisions by solving an inverse problem ................................................... 389 H.J .S. Dorren, R.K. Snieder One dimensional inverse scattering using data contaminated with errors ............................................. 405 N. Canosa, R. Rossignoli lnformation theory based methods for the reconstruction of quantum wave functiona .............................................. 412 A.G. Ramm Stability estimates in inverse scattering ............................... 419 M. Znojil Re-construction of polynomial potentiala with a perturbation interpolation constraint ................................. 458 A. V. Sarantsev Dispersion diagram method and deuteron as two nucleon composite system ........................................ 465 List of authors .......................................................... 481

An Overview of Some Inverse Problems and Techniques D.N. Ghosh Roy Sachs and Freeman, Landover, MD 20785, U.S.A.

1 Introduction There are numerous problems in physics the solutions of which require the inversion techniques used for reconstructing the potential in 1-D Schrodinger equation. An important generalization of this problem involves the nonlinear Schrodinger equation and its associated two-component scattering problem of Zakharov and Shabat [1). The connection between these inversion techniques and the physical problems to be solved is provided by the inverse scattering transforms [1-6). This article is a partial overview of the applications of the Zakharov-Shabat problem (ZSP), nonlinear Schrodinger equation (NLSE) and inverse scattering transform (IST) to certain problems in nonlinear optics. We begin with a discussion of the inverse scattering transform.

2 Inverse Scattering Transform (IST): A Brief Overview The 1-D Schrodinger equation and its generalization, the ZS-problem

u'(x, ..\) + i..\u(x, ..\) = q(x)v(x, ..\) v'(x, ..\)- i..\v(x, ..\) = r(x)u(x, ..\) with potentials q and r provide examples of a unity underlying seemingly contrasting physical situations. These equations, although describing linear scattering, play a fundamental role in the solutions of a large class of nonlinear partial differential evolution equations in physics which are described by equations of the form Yt(x,t) = Ny, subscript t denoting differentiation intime and Nisan operator acting on a sui table function space. The fundamental idea is to associate an appropriate linear potential scattering problem with the evolution equation to be solved. This is done via the Lax pair consisting of a spectral operator L and a time evolution operator M (see Section 5.1). The general form of M is given by the well known AKNS theory [6). The initial value y 0 = y(x, t = O) is substituted for the potential in the linear problem which is then solved forwardly to obtain the spectral transform S(..\, t = O) consisting of reflection data, bound

2

state eigenvalues and their normalization constants (analogous to the forward Fourier transform). S involves intime to S(.A, t) according to some specified and simple (which is crucial) rule. S(.A, t) is then backtransformed (an analogue of inverse Fourier transform) to the space-time domain by the Gel'fand-LevitanMarchenko (GLM) integral equation. The solution y(x, t) of the evolution equation is finally given by the solution of the GLM equation on the characteristic. IST is, therefore, considered to be the nonlinear analogue of Fourier transform

[6].

The general solution of a nonlinear evolution equation is a mixture of two modes: a continuous, nonlocalized radiation mode (the continuous spectrum of the eigenvalue problem) which decay in time and a superposition of non/inear normal modes or the solitons (the discrete spectrum) which are localized, permanent, stable waveforms that propagate intime by translation. This is equivalent to the principle of superposition in linear physics. These two modes are not discernible in the general solution because of their nonlinear mixing. They are, however, separated in the domain of the spectral transform. From another perspective, applying IST is equivalent to separating a Hamilton-J acobi problem into its action-angle variables by a canonica! transform [7].

3 Applications in Nonlinear Optics The ZS-system turns out tobe the linear scattering problem associated with the non/inear Schrodinger equation

Here, r = -q*. Together with the soli ton and the radiation modes, ZSP and NLSE enjoy wide applications in nonlinear optics. To overview some of these applications is the basic motivation of this paper. The major applications are in optical pulse propagation and wave-wave interactions and depend upon whether the optics is strongly or weakly nonlinear, that is, whether the light field is resonant or not with the quantum transitions of the optica! medium. In a weakly nonlinear process, the polarization bas a Taylor expansion

x(i) is the susceptibility of i-th order, EEE ... the tensor products of the fields and Maxwell's equation is written as

At resonance, the series expansion breaks down and the polarization cannot be separated into its linear and nonlinear parts in Maxwell's equation. This is the regime of strong nonlinearity. Optical pulse propagation is considered first.

3

4 Optica! Pulse Propagation Although nonlinear optics is replete with the propagation of pulses of numerous varieties, the pulses which are of interest here are solitons. As pointed out by Akhmanov [8], solitons in nonlinear optics are at least of three types: parametric, resonant and Schrodinger. A parametric soliton is generated in a three-wave process in a collinear, co-propagating geometry in the so-called given envelope approximation, that is, no pump depletion. The interacting wavepackets differ widely in intensity and frequency (usually a fundamental wavepacket and its second harmonic) and the soli ton formation is due to a balan ce between the effects of wave group delay and nonlinearity. The solitons may broaden as they propagate, but their functional shape remains unchanged. The analysis of these solitons does not necessarily require inverse techniques. Hence they are not considered here any further, details being available in [8, 9]. We, therefore, proceed to discuss the resonant solitons. 4.1 Resonant Solitons

This is the regime of strong nonlinearity and exotic phenomena occur in this regime. For example, if the pulse area exceeds a certain threshold value, the pulse may propagate as if in a transparent medium even though elementary theory predicts absorption. This is the famous ~elf-induced transparency or SIT, originally discovered by McCall and Hahn [10] in NMR. The medium absorbs in the leading edge and emits in the trailing edge of the pulse in such a way that no net energy is exchanged between the field and two-level medium in the process and the pulse propagates unchanged in shape. In practice, however, this applies only over a certain portion of the incident pulse. In another context, propagation of resonant solitons occurs in copropagating, forward stimulating Raman scattering (see Section 5.1 for definition of a Raman process). The analysis of soliton propagation under strong nonlinearity requires that the equations for the field, polarization and level population dynamics be solved self-consistently via optical Bloch equations [11, 12]. Only a cursory account of these processes and their connections with inv~rse scattering is presented here. IST solutions of SIT and propagation of short light pulses appear in [7, 13, 14]. In-depth studies of Raman solitons (including the effects of damping and analysis of integrability conditions) can be found in [15-20]. The fundamental equations governing the propagation of resonant solitons are the optical Bloch equations qz

uT W7

=
,

+ iilu = -qw,

=2(u*q+q*u),

u = u

+ iv,

il= w- WR T

=

detuning,

= t - x / c = retarded time.

The equations are written in a retarded coordinate frame and for pulse widths < < T1, T2, the longitudinal and the transverse relaxation time, respectively. For SIT, the level population difference w and the polarization components ( u, v)

4

form the diagonal and off-diagonal elements of the density matrix, respectively. q is the electric field multiplied by < /Jab > h, < /Jab > being the expectation value of the transition dipole moment matrix element between the levels a and b of the two-level system. < (J' > indicates that an averaging over the lineshape is required if the broadening is inhomogeneous. For Raman scattering, the time and space variables are reversed, (J' and w correspond to the pump and Stokes field, respectively, and q corresponds to optical phonons. The optical Bloch equations are solved by IST as follows. The L-eigenvalue problem is the ZS-problem with r = -q*. The M eigenvalue problem is given by the AKNS scheme, namely Uz

Vz

=A((, x, y)u + B((, x, y)u, = C((, x, y)u + D((, x, y)u.

The variable z can be either r or x. Then using the AKNS results and the theory of Gel'fand-Levitan-Marchenko, the potential is given by q = -2 K(z, z, t), where K is the solution of the GLM equation with the ZS-scattering data. The connection with IST can also be noted as follows. At exact resonance, for pulse widths < < relaxation times, and assuming that the energy stored by the oscillators is small compared to the energy in the field, the pulse area (Section 5.1) follows the McCall-Hahn area theorem [11] which obeys the sine-Gordon equation and this equation is exactly solvable by IST [12].

fz

4.2 Schrodinger Solitons Schrodinger solitons arise as a result of an exact balance between group velocity dispersion (GVD) and self-action of light when a pulse propagates in a transparent, dispersive and optically nonlinear medium under weak nonlinearity. Currently, these solitons enjoy a great popularity because of their potential applications in long distance· telecommunications by optical fibers. The formation of a Schrodinger soliton is shown in Figure 1 which is selfexplanatory. The picture is for the so-called bright solitons which propagate when the GVD coeflicient k2 (defined by k2 (â 2 kfâw 2 )w 0 ) is negative. If k 2 is positive, then the soliton is dark [21], that is, a permanent, stable, localized depression in the light level. In a dispersive medium, a propagating pulse acquires a linear frequency modulation (blue to red shift from the leading to the trailing edge, respectively). The self-action oflight, on the other hand, gives rise to self-modulation, that is, frequency modulation due to the intensity dependent refractive index (red to blue shift from the leading to the trailing edge, respectively). A soliton is formed when the characteristic lengths of these two countereffects exactly equal each other as indicated in Figure 1. Such a pictorial view elucidates the basic physics involved and also provides information about the integral characteristics of the pulse and the critical power needed for soliton formation. However, for answers to the important questions such as the shape of the steady-state pulse, the stability of the balance and pulse interactions, a deeper analysis is necessary. This is where the methodology of IST is indispensable. Moreover, for long distance propagation, accurate

=

5

tJT«t olJtll·iiCiioa 1SP~II

lc:ldiacedt~-....... tni6actdce

spre:adina

.y

(2.3)

16

the spectral matrix-function of which coincides with E(A). The corresponding potential p(x) is called the finite gap.

Remark 2.1: This class of potentiala coincides with potentiala constructed by the methods of algebraic geometry (Jacobi's inversion of hyperelliptic integrala) (see [3], [4], [5]). It coincides also with potentiala investigated by means of the theory of completely integrable Hamiltonian systems (see [6], [7], [8]). Remark 2.2: As is easily seen, if ali gaps collapse then

de ={21.,.-Jx,A>O' dA O , A~ O d7] dA= O,

d( = { 217f-/). ' A ~ O

dĂ

, A< O

O

These functiona are entries of the spectral matrix-function of the equation -y"

=

Ăy, -00

< xoo.

=

Therefore the finite gap potentiala can be considered as natural generalization of the trivial potential: Po( x) O. Together with the equation (2.3) consider for arbitrary t E R the equation

- y'' + q(x + t)y

= Ăy.

(2.4)

= p(x +t) is also a finite gap potential (n-gap) with the same spectrum as that of equation (2.3). Contrary, the spectral parameters ek(t) and ck(t) = ±, 1 ~ k ~nof equation {2.4) are moving with t. Put

It can be proven that Pt(x)

n

P(x, Ă) =

II (Ă- ek{x)), k=l

~

cj(x)J-R(ei(x))

Q(x,Ă)=P(x,A)~(Ă-'·(x))P'(A)l J=l

S(

.Eo(x, y; A) \

1\-

Z

M~(x, y; A)d>.Eo(x, y; A)

\

1\-

Z

(5.6)

(5.7)

The integral representation of an arbitrary function can be obtained by standard arguments from the integral representations (5.6) and (5. 7) Theorem 5.1: Let Eo(x, y; A) and M(x, y; A) have the same meaning as in Lemma 5.2. Let f(y) bea smooth function with bounded support. Then

f(x)

=

21: :. (l

M( x, y; ,\ )d,Eo(x, y;

A)) f(y)dy

(5.8)

22

Remark 5.2: The expansion formula (5.8) can be extended to the case of measurable functions without bounded support. But in every particular case we have to study this possibility (see the following section). Remark 5.3: Suppose that the Cauchy function has the form (5.5). The spectral function of a differential equation of second order can in many cases be written in the form

Eo(x, y; .A)=

1>.

00

..y,

O~

x

- y" + q(x)y = >..y,

O~

x~

~

71',

(8.1)

71',

(8.2)

both with Dirichlet boundary conditiona. Suppose that both operators have the same spectrum {>..n}, n 2: 1. Denote by oo, O :S t < oo, lxl < oo and satisfying the following analogue of the Sommerfeld radiation condition used in linear problems,

r~~ { ~; (x, t)- 4i [(1- c

2)

112

-

carccos c] Bn(x,

t)}

= O,

(22)

35

to be integrable. As we will see shortly the inverse problem attached with (4) brings out new features not present in standard P.D.E's. We believe that the Toda differential-delay equation is the first of what will be yet another new class of interesting nonlinear evolution equations to the class of nonlinear integrab1e equations solved by IST. As is common with IST integrability in 1+1 dimensions, we find that a suitable Riemann-Hilbert (RH) factorization problem plays a central role in the solution of this equation. In this case the RH problem requires factorization across an infinite number of strips along with a suitable Riemann surface which specifies the underlying branched structure. This Riemann surface is given by

(27) The inverse function l(k) has for its natural domain an infinitely sheeted Riemannsurface with branch cuts alongthe semilines {kR = mr,(-1)nk1 +(::; O, n E Z} where

( = Ln((J5 + 1)/2)- JS.

It can be proven that its branches can be labeled by an integer n; our convention is that (n- 1)7r O and of lo(k) for kR -A, and

,(t, X) ] e(t,x)= [ -'(-A-t,x)+1 ' where '(o, x) E L 2 {0, -A) and is otherwise arbitrary. In terms of ,(t, x) and P(k, x), from (4.10) and {4.16) we obtain m

k ikA rAd [ ,(t,x) ] ikt (k ,x ) =c() x P (k ,x ) { ie la t _,(-A-t,x) e

1 ]} + [ eikA ,

or equivalently m(k, x)

= c(x) P(k, x)

-ikt 1 ]} { ik larA dt ['(-A-t,x)] _,(t, x) e + [ eikA

o

(4.18)

If 1- H(x) has mixed sign, then one can modify the above procedure by using an analog of the method outlined in Section 10 of [8). As seen from (4015) and (4.18), the general solution of (4.1) by the above method is not unique; in order to solve the inverse scattering problem of the recovery of H(x), among all the solutions of {4.1), either we must pick the one that will give us the Faddeev functions of {1.1) or we need to find appropriate restrictions on the solutions of ( 4.1) so that they will lead to H ( x) Currently, we are working on this problem. Next we will illustrate the method outlined in this section on the example used in Section 3 with the scattering matrix given in (3.2). Corresponding to the potential Q(x) in (3.1), we have o

8 ra1{k) = [

k+i/2 - Văi/2 k+i/2] k+i k+i k-i/2 - Văi/2 k+i/2 k+i k+i

o

Solving the Riemann-Hilbert problem (4.3) by the method of [1,7), we obtain P(k, x)

= [ flie2ikx~3ik+Văik k+i

k22i] ' k+i

48

Thus from ( 4. 7) we have

c(x)

J3ex - 1

= J33ex + 1 ,

X~

0.

In this example, the Faddeev functions for (1.1) are given by 0,

X~

_ k + 2i J3ex- 1 -ik(1+3VS) 4ik/(VSe"'-1) mr (k ,x ) - k . ! O. One can therefore apply Cauchy theorem in the upper half-plane with cuts. This gives:

R(k ) = C(k ) ,q

,q

_1_1

F(k) + Ynpd q - Z·11: + 271' Z·

_

R(k, x)

00 00

q-

X -

·o dx,

(3)

Z

where C(k, q) is the cuts contribution, F(k) is the deuteron charge form factor,

Ynpd=-~

(resiS(q)'

(4)

.

q=u;

and S( q) is the amplitude of np-scattering in the 3 8 1-state. By taking the real part of Eq. (3) and using the relations lm B(k, q) =O,

= Im M(k, -q) = M(k, q) exp(- i o(q)) sin6(q),

Im M(k, q)

where o(q) is 3 8 1 -phase shift, one can rewrite Eq.(3) as follows:

M(k, q) = B(k, q) + 2ReC(k, q) +

+.!_1 71'

00

-oo

2q Ynpd F(k) 2

q

+~~:

2

+

M(k, x) exp(- i o~x)) sin 6(x) dx. X - q - ZO

(5)

The solution of this equation is customary used in the form:

M(k,q)=B(k,q)+2ReC(k,q)+ - il(q) 71'

2q Ynpd F(k) 2

q

+~~:

2

-

1oo ImG-1(~~ (B(k, x) + 2ReC(k, x) + 2gn~dx F~k)) dx, q+ _ 00 X -

Z

where

.O(q) = exp

X

{.!.joo 71'

6(x) q-

_ 00 X-

·o dx}.

Z

11:

(6)

53

However, since 8(+0)

= 1r,

-1 P 71"

1

00

8(-0)

= -71"

1

we have

8(x) - dx . . . , -2 log(q) q

-oo X -

as q --+ oo. Correspondingly, .G(q) . . . , q- 2 and M(k, q) has a pole at q =O unless the following sum rule holds:

1

oo -oo

Im .a:l(k) (B(k, q) + 2 Re C(k, q) + q

2gn~d~ F2(k)) dq =O' q

(7)

K

where n = 1 or 2. Now we shall see that M(k, q) with .G(q) chosen in any way has a meaningless pole at some point of the complex plane. For this reason the sum rule (7) has to hold in any dispersion calculation of M(k, q). Following the Carleman's method (see e.g. Ref. [10]) we introduce the function __1_1 00 M(k,x) exp(-i8(x)) sin8(x) d (k (8) X. F 'z ) - 271" Z. _ X- Z 00 It is analytic at all points z not on the real axis and tends to zero as z By Eq.(5) and the Plemelj formula _1

1

00

271" i _ 00

f(x) dx X-

Y- i 0

1 __

the boundary values of F(k, z) as z

271" i

--+

1

00

_ 00

f(x) dx X-

_ f

Y+ i0 -

--+

oo.

(y) '

q from above and below are such that

F(k, q + i O)- S(q) F(k, q- i O) = exp( i 8(q)) sin 8(q) G(k, q),

(9)

where S(q) = exp( i 8(q)) and

G(k,q)=B(k,q)+2ReC(k,q)+

2q 9npd F(k) · 2 2 q +K

Now to solve eq. (5) one has to find a function F(k, z) analytic for all points --+ q from above and below such that:

z not on the real axis with limiting values as z

(1) Eq.(9) holds for all points on the real axis;

(2)

F(k,z)--+ O at

z--+ oo.

We first find a function .G(z), such that

.G(q + iO) = S(q) .G(q- iO),

-oo

< q < oo,

(10)

with nonzero .G(q + iO) and .G(q- iO) . Does there exist the solution .G(z) of the boundary value problem (10) with no zeros and poles in the complex plane ? To answer this question one has to examine the index of S(q) , i.e. 1/271" times

54

tbe cbange in tbe pbase of S(q) resulting from a complete traversa! of tbe real axis [10]. In our case tbe index of S(q) equals -2. Eq. (10) gives: Ind .G(q + i O)- Ind .G(q- i O)= Ind S(q) By virtue oftbe logaritbmic residue tbeorem [10] Ind .G(q ± i O) = ± (N±- P±), wbere N± is tbe number ofzeros and P± is tbat of poles of tbe function .G(z) in tbe domain ± Im z > O, eacb zero or pole being counted a number of times equal to its multiplicity. Since Ind S(q) = -2, we bave (N+ + N_)- (P+ + P_) = -2, tbat is .G(z) bas at least two poles. We define tbe canonica! solution .Gc(z) as

.Gc(z) = ( - z.) exp z +z 2

(·.!.joo 7r

~z dx).

_ 00 X -

Tben tbe most general form of .G( z) is:

P(z) .G(z) = Q(z) .Gc(z), wbere P(z) and Q(z) are polynomials. By Eq.(10) one can rewrite Eq.(9) as

F(k,q+iO) -H(k q + z·o

wbere

H(k,z)

·o)- F(k,q-iO) -H(k _'O) ,q+z ,q z ' q- 2·o

= -2 1 . loo 1r2

_ 00

G(k, :c)

(11)

d:c

.G(:c+z' O:c-z )-.

Botb sides of Eq.(ll) are tbe brancbes of a function, say, A(k, z) , analytic everywbere except tbe points Pl, · · · , Pn wbere P( z) = O . If tbe degree of Q( z) exceeds tbat of P(z), A(k, z) bas also tbe pole at infinity. By tbe Liouville's tbeorem:

A(k, z) = H(k, z) + Co,o(k) + Co,l(k) z + · · · + Co,n 0 (k) zno +

+ ~ (C;,1(k) + ... + C;,n(k) ) ţ..J

J=l

z- p·J

(z- p·)ni J

Tberefore

F(k, z) = .G(z) [(H(k, z) + Co,o(k) + Co,1(k) z + · · · + Co,n 0 (k) zno+

+ ~ ( C;,1(k) + ... + {;;:

Z-

Pi

C;,ni(k) )]

(z- P;)n 1

Tbe function F(k, z) given by tbis equation bas tbe second order pole at z = -i ~ and tbe poles at tbe points q1 , · · ·, q,, wbere Q(z) =O. If tbe degree of

55

P(z) exceeds that of Q(z), F(k, z) has also the pole at infinity. Therefore F(k, z) will correspond via Eq.(8) to an amplitude M(k, q) if and only if the function H(k, z) + Co,o(k)

+ Co,l(k) z + · · · + Co,n (k) zno + + Cj,1(k) + ... + Ci,n;(k) )

f(

i=l

0

z- Pi

(z- Pj)n;

and the sui table number of its derivatives vanish at the poles. This gives rise to the N + 2 equations, N being the greatest degree of the polynomials P( z) and Q(z). These equations need be used to determine the functions C(k). As the number ofthe functions is N ,the two equations remain unemployed. They are equivalent to Eq. (7) with n = 1, or 2. Thus, Eq.(7) gives the necessary and sufficient condition for the solution of Eq.(5) to exist. Since the solution does exist, Eq.(7) must be valid. Let us study its consequences. If n = 1, the equation holds simply for the functions ImQ- 1 (q), B(k, q) and ReC(k,q) are odd in q. If n 2, Eq.(7) gives:

=

1oo Im.f.?: (q) (B(k,q)+2ReC(k,q))dq -oo q 1

F(k) = x:;Q(ix:) 7C' 9npd

(12)

Using this equation one can exclude F(k) from Eq.(6). This gives:

1 () 7C' 1 q

1

M(k, q) = B(k, q) + 2 ReC(k, q)Imf(x) . (B(k,x)+2ReC(k,x))dx, X - q- Z 0

00

_ 00

(13)

=

where f(q) (q 2 + x: 2) q- 2 Q- 1(q). Eqs.(12) and (13) indicate that B(k, q) and C(k, q) determine M(k, q), whereas F(k) equals g;;;d times the residue of M(k, q) at q = i x: . If so, the function C(k, q) must be chosen in such a way as to remove the meaningless poles of the amplitude. To this end it is necessary to make Eq.(12) valid. Let us show that it ,is impossible to do when C(k, q) =O . Indeed, if C(k, q) =O, Eqs.(12) and (1) give:

Im.fi?-l(q) ) sin(qk)dq 1oo ( u(r)- x:22 Q(ix:) 1oo -oo q O

Therefore

2

7C'9npd

x:2 Q( ix:)

u (r) = 2

7C' 9npd

kr io(2 )u(r)dr=0.

1oo Im Q-1 ( q) sm. (qr) dq . -oo q 2

However this equation cannot be valid since otherwise the normalization integral of u(r) would be markedly less than unity. For example, if one takes Q(q) in the effective range approximation, then

1

00

u 2 (r) dr

~ 0.75.

56

The more accurate approximation for il(q) increases this value by about ten percent only. Thus Eq.(12) indicates that if C(k, q) = O, the dispersion theory contradicts with the description by the wave function.

3 The unphysical cuts contribution and the neutron-proton wave function Since the unphysical cuts are proved to be important, one has to develop a method for taking them into account. As we pointed out in the Introduction, the wave function has also to be shown to exist, that reproduce the results of the dispersion theory. In this Section we shall find it explicitly. We define B_ ( k) as

t" lo

B_(k) =

i

= -k

exp(-ikr)jo(kr)u(r)dr =

1"",.

where 1

g(a) = -2 . 1f Z

2 q - k /2 - ia . da, g(a) log q+ k/ 2 -w

(14)

lioo exp(ar) u(r) dr. -ioo

V.de Alfaro and C.Rossetti [9) found the unphysical singularities to lie at a distance of J.l just above the branch points of B_ ( k, q) , J.l being the pion mass. Therefore it is natural to put

C(k, q)

~-ii""

B_(k, q- ix) h(x) dx.

(15)

The standard imitations of the unphysical cuts contribution by a sum of logarithms or poles are obtained from this equation if h( x) = L: Aj 8( x - ai) or h( x) = L: Aj 8' ( x - ai) respectively. z.From Eqs.(14),{15) it follows that 2ReC(k,q)

~

-21

where

h(r) =

00

sin(qr)h(r)j0 (k;)u(r)dr,

1"" h(x)

exp( -xr) dx.

By inserting Eq.{16) into Eq.{13) we get:

M(k, q) =

- -17r f(q)

1""

-00

1""

{1- 2h(r))( sin{qr)-

. (kr) u (r ) d r lmf(x) sin(kx) d x ) Jo. 2 X - q- iO

(16)

57

the comparison of this equation with Eq.(1) indicates that the dispersion theory is equivalent to describing the np-system by the wave function:

1/J(qr) = (1- 2h(r)) ( sin(qr)-

7r

;(q)

1:

Im;~~ :ni~r) dx)

.

However, as q --+ oo this function tends to ( 1 - 2 h( r)) sin( qr) instead of sin( qr) Therefore simple pole or logarithmic approximation for C(k, q) and their generalization given by Eq.(15) do not give a satisfactory description of the np-system lnstead of Eq.(15) we consider

C(k ) ~ Co(k, q)- Co(k, ib) 'q 1 + iqfb '

(17)

where Co(k, q) is given by Eq.(15) and b is an arbitrary parameter. From Eqs.(14) and (15) one gets:

11

C(k, q) ~ -

00

1

1

00

dx h(x) 1 . /b + aq l q-k/2-i(a+x) _ -k/2-i(a+x-b)) 1 ( og q + k/2- i(a + x) og k/2- i(a + x- b) · q

"

g(a) da

IA

This form of C(k, q) with g(a) = N( 6(a- x)- 6(a- b)) and h(x)::::: v 6(x -1') was used by D.Braess [11]. With C(k, q) given by Eq.(17) the results of the dispersion theory are reproduced by the wave function:

1

1/J(q, r) = sin(qr) + a(q, r)-

1 Imf(x) . - ! q() _ x - q - a.0 ( sm( xr) + a( x, r)) dx , 7r 00 00

(18)

where

loo h(x)( exp( -xr- iqr)- exp(br- xr)) dx

a(q, r) = 2 Re 1 +iiqfb

and b < p. • The most general form of a( q, r) is: a(q,r)=2Re

1oo dx 1oo dy q-h(x:/y)2 (exp(-xr-iqr)IA

IA

ay

-exp(yr/2-xr)).

(19)

It corresponds to

C(k, q) = -i

1oo dx 1x dyh(x, y) B_(k, q- ix)~~~k, iy/2- ix) . qIA

IA

ay

We can use for h(x, y) a phenomenological expression with a number of parameters. Since h(x, y) enters the expression for the wave function, we can use

58

the normalization condition of the wave function to fix the parameters involved in h(x, y).

1

00

.,P*(q, r) .,P(q'' r) dr =

i o(q- q')

(20)

4 The wave function construction. Let us demonstrate now the method formulated above using as an example a simple case when the scattering phase is given by effective radius approximation:

(21) We suppose that a> O, so that there is one bound state. In this case one must add the factor k 2~K. 2 in the right hand side of Eq. (6) while finding Q(k). Now

Q(k) = k + ib k- ib' where K.

1[

~] 1 - ---;- ' = ro - 1 + V

(22)

1 [1 + b = ro

V~] 1 - ---;- .

(23)

Let us approximate the function h(x, y) in Eq.(19) by

h(x, y)

=V

(24)

o(x- y)o(x- 2v),

where v and v are the parameters satisfying the condition 2v > J1. ~ 0.707/m- 1 . We find from (19) and (24) that e-(ik+2v)r - e-vr

a(k,r)=vRe

.,P(k, r) where

k

(25)

. - zv

= 1/Jo(k, r) + v tPt(k, r),

(26)

. k(b + K.) eikr _ e-br 1/Jo(k,r)=smkr+ k - ZK. . · k - Z'b ,

t/J1(kr) = Re k(b + K.) (k-iK-)(k-ib)

e-(2v+ik)r- e-vr

k

.

-w

(27)

+

{ .e-(2v-ik)r _ e-vr

z

k+iv e-(2v+b)r - e-vr }

b+v

·

(28)

The physical meaning of this separation (26) is obvious: the function .,P0 is the wave function arising in the dispersion approach in the case when the unphysical cuts are neglected and tPl contains the contribution of these cuts.

59

It is easy to verify that no 1/Jo nor 1jJ1 do not satisfy (20) exactly. We can however vary v and v in such a way that the difference between l.h.s. and r .h.s. of (20) for 1/J would be much less than for 1/Jo. In other words, denoting these differencies by d and d0 , respectively, we shalllook for such v and v that

lld(k, k')W lldo(k, k')ll 2 0 (p,-k+~o:)e(u,o:)-k + zu/2

4> 0 (p, k + io:) e(u, o:) S(k)] iwL/2 k

+ iu/2

+~ F~(k,p)

e

+

[1- S(k)] eiwL/ 2 ,

{43)

where {44)

(45) and finally

4>o(

) _ ((2J -l)!!p-J UJ-l(pr)

p,r -

O

O

{2J +3)!!p- 1 - 2 uJ+l(pr)

)

{46)

Here UJ±l(pr) is the Riccati- Beaael function. Taking into account the Eqa.(43- 46) one can show that .,pint(k,p) has the following aingularitiea in the upper half-plane: 1. The deuteron pole at the point k = i~~: originated from the pole of S(k).

62

2. Branching points of matrix ~ 0 (p, -k + io:) elements at k = ±p + io: (

ţt ~

0:


0). So one can use Cauchy theorem and obtain the following dispersion relation

.,p+(k,p) = .,po(k,p) + ~ 11"

1oo dk I~.,P+(k,~) + k - k- zO ...:. 00

+2Re A(k, p) + 2Re res lk=iK.·.'.,pint(k,p)

(47)

with A(k,p) denoting the unphysical cuts contribution. The dispersion relation (47) is satisfied by the matrix wave function of the np-system in the superposition of 3 81 and 3 D1 statea. Let us transform the dispersion relation (47) into the equation for the function .,p+(k,p). For this purpose we can express Im.,P+(k,p) in terms of .,p+(k,p) it self: 1- s(k) Im.,p+(k,p) = .,p+(k,p) (48) 2i We use the simplified notation:

Now we can rewrite Eq.(47) in the form

+

_

.,p (k,p)- B(k,p) +

~ 11"

1oo dk,(k'.,p+(k',p) 1- s*(k') k- iO) 2i ' -oo

(50)

where

B(k,p) = .,P 0 (k,p) + 2ReA(k,p) + . +2Re k-1 iK res 1k=i" .,pmt(k,p)

(51)

Let us use Eq.(50) to present .,p+ in terms B and S. We introduce the matrixfunction dk' .,p+(k,p) . 1- S*(k') ( ) = _1 (52)

O "p;,"(k,r)"p,,,"(k',r)dr=~Oli'O(k-k'). 2

(68)

We are looking for such values of parameters that the Eq.(68) is fulfilled with essentially greater accuracy than at a(k, r) =O. In conclusion, we have shown that if the deuteron electrodisintegration amplitude satisfies Mandelstam representation, then the np-system wave function has a concrete form and can be expressed in terms of the np-scattering phases. This wave function satisfies a variant of the dispersion relation (50). The dispersion relation (50) is valid for a class of potentiala much larger than that having the form (35). So the dispersion relation can be used to find the explicit realistic wave functiona for two nucleons systems. The behavior of deuteron wave function u(r) is similar Paris wave function (see [8]). I thank Dr. A. F. Krutov for helpful discussion.

References 1. K. Chadan and P. C. Sabatier, Inverse Problem in Quantum Scattering The-

ory, 2nd ed. (Springer, New York, 1989). 2. R. G. Newton, Inverse Schrodinger Scattering in Three Dimensions, (Springer, New York, 1989). 3. R. G. Newton, Theory of Waves and Particles, 2nd ed. (Springer, New York, 1982). 4. Th. Kirst, K. Amos, L. Berge, M. Coz and H. V. von Geramb, Phys.Rev. C 43(1989)912. 5. A. I. Kirillov and V. E. Troitsky, Yad.Fiz. 27(1977)288 (Sov.J.Nucl.Phys. 25(1977)157). 6. V. E. Troitsky, Yad.Fiz. 29(1979)23~~ 7. V. M. Muzafarov and V. E. Troitsky, Pis'ma Zh.Eksp.Theor.Fiz. 30(1979)78 (JETP Lett. 30(1979)70). 8. V. M. Muzafarov and V. E. Troitsky, Yad.Fiz. 33(1981)1461. 9. V. de Alfaro and C. Rossetty, Nuov.Cim. 18(1960)783. 10. G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variables, (Mc Graw-Hill Book Company, New York, 1966); F. D. Gakhov, Boundary Value Problems, (Pergamon Press, Oxford, 1966).

66

11. D. Braess, Z.Phys. 184(1965)241. 12. R.Vinh Mau, A.Martin, Nuov.Cim. 20(1961)390.

M ultidimensional and Three-Body Inverse Scattering Problems in the Adiabatic Representation A.A. Suzko Laboratory of Theoretical Physics, Joint Institute for Nuclear Research (Dubna), Head Post Oftice, P.O.Box 79, Moscow 101 000, Russia

Abstract. In the adiabatic representation, the multidimensional and three-body inverse scattering problems are discussed on the hasis of consistent formulation of both the multichannel inverse problem for gauge systems of equations describing "slow" dynamics ofthe system, and the parametric one for "fast" dynamics. The method of constructing a wide class of exactly solvable multidimensional models is investigated by comparing the Bargmann potentials with the parametric family of inverse problems and systems of equations with covariant derivatives. A problem introducing an extra matrix of scalar potentials so as to conserve supersymmetry and thus conditions for topologica! effects is studied. A direct generalization of the Witten supersymmetric quantum mechanics for gauge equations with additional scalar potentials is given. Coupling of supersymmetry and geometric phases and the influence of additional scalar potentials under the degeneracy of the ground state, and as a result under topologica! effects, are discussed. Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schroedinger ones with an additional functional dependence (h(r)) in the right-hand side of equations are constructed. Keywords: Inverse scattering, adiabatic representation, supersymmetry, geometric phases.

1 Introduction We think that the most promising method for solving complicated multidimensional, many-body and few-body problems in quantum mechanics is the adiabatic representation method [1], [2] that allows us to take into account the mutual influence of slowly changing externa! and rapidly changing interna! fields. Correspondingly, one introduces slow "s" and fast "f" degrees of freedom for slow and fast subsystems connected with each other. The total Hamiltonian is accordingly decomposed into two parts, H = h3 + hf. The fast subsystems can be considered to be embedded in the slow subsystems which in turn affect the properties of the fast subsystems. In this approach, solution of the whole scattering problem is reduced to two effective scattering problems in the spaces of a lower dimension, then the original

68

M = B x M. One is the parametric scattering prohlem for the fast Hamiltonians hf (X) parametrically dependent on slow coordinate variahles X, helonging to some manifold B: X E B. Another prohlem is the effective scattering theory for the system of gauge equations with induced gauge fields, generated hy the procedure of adiahatic expansion of the total wave function lli(X) = "L,f through the unitary bilocal operator

U(X, Xa) =: U(X)

I=le> U(X, X a) , U(X, X a)=< ejci>(X; .) >,

(13)

that accomplishes a translation of the frame from X o to X. From the definition of the operator A (6) with (13) and (4) we get

(14) and the covariant derivative (11) associated with the gauge field Av can be rewritten in terms of the operator U(X)

(15) Now, by using the unitary gauge transformation

J Xo

U(X,Xa) = Pexpi

A(X')dX'.

X

(16)

73

obtained from (14) and (13) under the condition (12) we can annihilate A and reduce the system of Eqs.(5) coupled through the kinetic energy operator to a system of ordinary equations coupled through the effective potential matrix {-

d~ 2 + U(X)U(X)U- 1 (X)- P

2)

}x'(X, P) =O,

(17)

for new coefficients x' connected with x by (8). Note, that in the one-dimensional Abelian case path ordering in expression (16) is unnecessary. As is well known from vector analysis, if the curl of the vector potential (6) vanishes at all X, 'V x A = O, we can eliminate the gauge vector potential by a phase transformation. In the adiabatic representation, vanishing of the matrix tensor RJ.Iv = âJ.IAv - âvAJ.I - ig[Av, AJ.I] is equivalent to this requirement, because we have non-Abelian gauge fields. Proceeding with the transformation (8) where U(X) ( 16) and using ( 15) we find that the gauge vector potential vanishes. If the vector potential contains singularities, then the vanishing of curvature RJ.Iv is not enough for eliminating the vector potential at all X. Note, more interesting effects take place when RJ.Iv f= O. These phenomena are connected with Berry's opening of geometric phases in simple quantum systems [11]. Berry demonstrated the existence of magnetic monopole fields in dynamical systems, which arise naturally in a gauge theory framework [27]. If we want to evaluate the integral (16) over a closed loop in R-space, then it is better to convert the line integral by a Stokes theorem to a surface integral over the "magnetic field" B = 'V x A (in our case B = RJ.Iv ). Then the line integral over a closed loop C becomes (18) and because rot grad = O it does not depend on the gauge transformation A -+ A- 'V x. When eigenvalues (potential curves) intersect at some point of R-space, then the vector potential A(X) is singular and closed loop gives a non-zero result. We discuss this problem below. Here we assume that the potential curves are well separated and not degenerate. 2.2 The inverse problem for a system of equations describing slow dynamics. The inverse scattering problem for the system of gauge Eqs.(5) consists of severa! steps: the determination of the S-matrix in terms of known multidimensional amplitudes and subsequent reconstruction of effective vector, A, and scalar, V, potential matrices, as well as solutions. It is solved by using a unitary transformation of the gauge type (13), (16) that reduces the system of Eqs.{5) to a system of ordinary equations coupled through the effective potential matrix (17). After that we may apply the standard methods of the multichannel inverse problem [28]-[32] to the system of Eqs. {17), provided that we know the corresponding

74

scattering matrix S' (P) and information on states of the discrete spectrum, their positions {EA} and normalizations {Mi}. lfthe gauge transformation is unitary (12), then it turns out that

Mi

= MĂ '

S'(P)

= S(P) '

:F'(P)

= :F(P).

(19)

lndeed [3, 7], substituting the relationship (8) for Jost matrix solutions F' with F in the standard definitions of normalization matrices Mi and the symmetric S'matrix S'(P) = p- 112 P_(P)(:Ff.(P))P 112 and using the unitary of matrices U(X), we get (19). The S-matrix corresponding to the describing system (5) is defined as

(20) where the Jost matrix functiona :F±(P) by definition is equal to

:F±(P) = Wd{F±(X, P),tP(X, P)}

= F±(X, P)M>(X, P)- DF±(X, P)tP(X, P).

Here tP are matrices of regular solutions to the system of Eqs.(5), the tilde means transposition. Until now we have assumed that there are no level crossings or quasi-crossing of levels fn(X) responsible for the violation of unitary U at these points and the appearance ofsingularities in (6), (5) and owing to this nontrivial geometric phases. Let us reproduce the potential matrix Uf'(X) = U(X)hf(X)U- 1 (X) from the scattering data {S(P), "'Ă• MĂ} and determine solutions x' of the system (17) using the generalized multichannel Gel'fand-Levitan-Marchenko formulae [33, 34]

J

oo(X)

K(X, X')+ Q(X, X')+

K(X, t)Q(t, X')dt = O,

{21)

X(O)

U'(X) =U' (X) =f 2 d~ K(X, X),

(22)

J

(23)

oo(X)

F'(X, P) =F' (X, P) + o

o K(X, X') F' (X', P)dX'.

X(O)

Here the integration from X to oo with the minus sign and the integration from O to X with the plus sign corresponds to the Marchenko [29, 30] and Gel'fandLevitan formulation [28], respectively. The multichannel system of Eqs.(21) has tobe solved with respect to the kernel of a generalized shift K(X, X') on known kernel Q(X, X'). For example in the Marchenko approach, the kernel Q(X, X') A

o

o

0

is determined from scattering data S'(P), {Mi}, {EA} and S'(P), {M'A}, {E},

75 o

corresponding to the system of Eqs.(17) with potential matricea U'(X) and U' (X), respectively

J F' ()()

Q(X,X') = ..!_ 211"

+ L: .N

A

-

(X,P)[S'(P)- S'(P)]F'(X',P)dP

-oo

0 F' (X, iK:A)M~F'(X', iK:A)0

A

o

0

L: F' (X, i K:A) M\ F'(X', i K:A)· N

o

0

o

0

(24)

A

o

o

The Jost solutions F' (X, iK:A) and F' (X, i ~A) of system (17) with the matrix o

o

U' (X) should be taken at the energies of the bound states EA and EA, of o

both problema with U'(X) and U' (X), respectively. In the Gel'fand-Levitan approach with account :F'(P) :F(P), by analogy (19), we get p'(P) p(P) for spectral matricea and o~= OA for normalization matricea. Proceeding with the inverse transformation (16), (8) we get searched scalar and vector potentiala by formula (9) and (14). For example, for the case with

=

=

o

V= O and real hasis functiona after solution of the inverse problem (21)-(23), we can find terms E(X) by solving the algebraic problem for eigenvalues [7], [8]. Actually, using the diagonalized procedure

U'(X)G(X) = G(X)U(X) = G(X)E(X)

(25)

we find eigenvalues E(X) VX E B but the matrix of diagonalization G(X) is defined ambiguously. Note, it is connected with gauge transformation U(X). In general case the Eq.(5) is covariant under U(1) x SU(N) transformation at each X. Therefore we have following properties detU 1, trU' 1. First, we shall analyse the more simple situation when the potential VI (X) is known. In a three-body problem, this potential may bea sum oftwo-body potentiala, Vf(X) =:Ea Va(X). We should determine an additional potential V'(X). In the example under consideration, V"(X) = V123(X), a three-particle interaction potential. For every fuced value of the slow variables X (for example radius), we determine the reference functiona (k, y) 4> (k, y')d[p(x; k)- p (k))

o

+ L c~(x) 4> (iKn(x), y) m

o

o

o

4> (iKn(x), y')-

L m

o2 o

o

o

o

cni/J (i Kn, y) 4> (i Kn, y') (62)

n

n

we get

KGL(x; y, y') = -c 2 (x)cjJ(i~~:(x), y) sinh[~~:(x)y'J/~~:(x).

(64)

The normalization c~ (x) of the potential curves En (x), as usually, we define by regular solutions 1/J(x;k,y): c~(x) =

00

{J lc/J(iKn(x),y)l 2 dy}- 1 .

The regular o solution of Eq.(48) at k = iK(x), the corresponding potential curve, is

.

K(x) sinh(~~:(x)y)

1/>(z~~:(x), y) = ~~: 2 (x) + (c 2 (x)/2)[sinh(2~~:(x)y)/2~~:(x))- y).labe/1. 86

(65 )

At last, taking an account (??) in eq.(64) and substituting the result in the o formulas (58), (57) at V (y) = O, we obtain the analytical relations for the twodimension potential V(x; y) and pertinent solutions with parametric dependence on x:

(66)

r

sinh[K(x)y')sin(ky')dy' A..( ·k ) = sinky _ 2( )A..(" ( ) ) K(x)k C X '+' ZK X 'y lo k X, 'y

'+'

(67)

The method of construction may be verified as follows. Taking the limit in (67) when y--+ oo and separating the terms with exp(±iky) we determine that their

85

coefficients are just the Jost functions f±(x;k) coincident with (60). Using them and definition (49) we construct the parametric S-matrix

s(x; k)

= f-(x; k) = (k + ix:(x))

2

(k- ix:(x))2

f+(x; k)

.

o

For m bound states and reference potential V (y) ::j; O, 0 IIm k- ix:n(x) f+(x; k) =f (k) n k + ix:n(x)'

(68)

where potential curves &n(x) = -x:~(x) are obtained by solving the multichannel system of equations in slow variables and subsequent diagonalization (25). It is easy to write an m-term generalization offormulae (63)-(66). The kernels ofthe main Gelfand-Levitan Eq.(56) are written as follows

n m

o

KGL(x;y,y') =- Ec;(x)l,il(ix:n(x),y) l/l [ix:n(x),y'].

(70)

n

Then, using one of the relations of the inverse scattering problem, (56) or (58), we derive the solutions l/l[ix:n(x), y] for bound states &n(x) = -x:~(x), dependent on the dynamic variable x as a parameter,

l/l[ix:n(x),y] =

m

o

E l/l [ix:;(x),y]P;-;/(x;y),

(71)

j

where

Pn;(x; y) = Onj

+ c;

1Y ~ [ix:n(x), y'] ~ [ix:;(x), y']dy'.

Inserting the expression (71) obtained for l/J[ix:n(x), y] into formula ( 70) for the kernel KGL(x; y, y') and using the relationships of the parametric inverse problem (56)-(58) we obtain a closed analytic form for the potential and solutions

d2 V(x; y) = -2 dy 2 lndet IIPn;(x; Y)ll;

(72)

o

l,il(x; k, y) =l/l (k, y)

[Y

- LL l/l [ix:;(x),y]Pi~1 (x;y) Jo m

m

n

j

o

o

o

l/l [ix:n(x),y'] l/J (k,y')dy'.

(73)

O

The procedure is easily realized in the spherical coordinates, with the angle taken as a fast variable; and the coordinate, as a slow variable, and vice versa.

86

The Marchenko approach Reflectionless (transparent) potentials along the fast variable describe the one-dimensional inverse problem along the whole axis with the zeroth reflection coefficient, sref = O. The transmission coefficient str with the absolute value equal to unity is a rational fraction. It should be noted that various versions are possible: transparent potentials along both the slow and fast variables, transparent ones along one of them, cutoff potentials along one of the coordinates or along both, and so on. The inverse problem on the line [44]-[46] is very similar to the two-channel problem with two decoupled integral equations. However, since V(x, y) is expressed both through the kernel of one equation, K 1 (x; y, y'), and that ofthe second equation, K 2 (x; y, y'), it is sufficient to find one Ki(x; y, y') by a formula, coincident with (56). Then QM (x; y, y'), o

defined in this case at V (y)

=Oby the formula .

Joo sref(x;k)exp[ik(y+y')]dk -oo + L 'Y~(x) exp[-Kn(x)(y + y')],

QM(x;y,y') = -21

7r

m

(74)

n

will contain only the contribution of states of the discrete spectrum m

QM (x; y, y') =

L 'Y~(x)exp[-Kn(x)(y + y')].

(75)

n

Analogously, for J{M (x; y, y') we have

L ~~(x)f(iKn(x), y) exp[-Kn(x)y']. m

J{M (x; y, y') =-

(76)

n

For the Jost solutions at k = iKn(x) we get from (56) the following system of algebraic equations m

f(iKn(x), y) =

L exp[-Kj(x)y]Pj~1 (x; y)

(77)

j

with the matrix of coefficients Pjn(x; y), parametrically (through the spectral parameters) dependent on x:

p. ( . ) _ Jn

X,

Y -

{J .

nJ

+

~~(x)exp[-(Kn(x) + Kj(x))y] · Kn (X ) + KJ·( X )

Upon substituting f(iKn(x),y) into KM(x;y,y') (76) and using (57), (58), we get d2

V(x; y) = -2 dy 2 lndet IIPnj(x; y)ll,

(78)

87

f±(x; k, y) = exp(±iky)

+

k)y] L 'Yn(x) exp[-x:n(x)y]Pnj (x; y) exp[{-x:j(x)±i ( ) "k . (79) _1

2

.

Kj X

nJ

Tz

Now using the potential curves and their normalizations with the parametric dependence on slow variables x determined upon solving the inverse problem for the slow system of equations and diagonalization procedure (25) we can reproduce the model multidimensional potential in an explicit form and get the corresponding solutions. In particular, in the considered case of two potential curves we substitute relations (47) and (51) into (78) and (79) and obtain twodimensional exact models in the closed analytic form. Now it is evident how one can construct many other exact solvable models not only two-dimension ones. For the case of one bound state we obtain a potential of the Ekkart type V

.

_

2

2x:(x)'Y 2 (x) exp[-2x:(x)y]

(80)

(x, y)-- 1 + (-y 2 (x)j2x:(x)) exp( -2x:(x)y)'

that is transformed to a simpler form if use is made of the substitution

and

{1 + exp(-2x:(x)(y - Yo(x)]} 2 = 4cosh 2 [x:(x)(y- Yo(x)] · exp(-2x:(x)(y - Yo(x)], V(x;y) =-

x;2( X) cosh 2 [x:(x)(y- Yo(x)]

(81)

.

The corresponding Jost solutions are written in an explicit form on the curves of bound states -x: 2 (x ), and at arbitrary values of k, respectively

f .

exp( -x:(x)y) (zx:(x), y) = exp[-2x:(x)(y - Yo(x)]'

(82)

. { exp(-2x:(x)(y - Yo(x)] } f±(x;k,y)=ex p(±zky) 1- 1 + exp [- 2x: ()( x y - Yo ()]() x x: x =f z"k).

(83)

Consider a more general case than (68) where the parametric Jost function (52) has N simple poles at points k if3i(x) and N simple zeros at k iai(x). This takes place in the radial or half-axis problems. This example permits us to demonstrate opportunities for construction of a parametric family of phaseequivalent potentials depending on N normalizing constants M:î (x) which are in turn functions of an extra dynamic parameter x. The dependence on the slow coordinate x is given by the operator of translation U ( x, ~) moving the frame ltiJ > from :FoX to Fc· In the scattering matrix corresponding to the gauge Eq.(5), poles can arise. They are manifested as geometric phases connected with a specific behavior of the connection A induced by functions of the hasis parametric equation.

=

=

88

For the parametric inverse problem, either radial or on a semi-axis, with the Jost function (52) generalizing (68)

!( . k) =liN k- iKn(x) k + ibn(x)' x,

(84)

n

the scattering S-matrix is written in the form

S(x; k)

k + ibn(x). =ÎI kk-_ţ iKn(x) ibn(x) k- iKn(x)

(85)

n

The kernel of the basic integral equation in the Marchenko approach (56) 00

;7r

Q(x;y,y')=

J[l-S(x;k)]exp[ik(y+y')] -oo N

+L

M~(x) exp[-Kn(x)(y + y')]

(86)

n

with (85) is rewritten as follows N

Q(x; t

= y + y') =-iL Res S(k = ibn(x)) exp[-bn(x)t] n N

+ L{-iRes S(k = iKn(x))exp[-Kn(x)t] n

+ M~ exp( -Kn(x)t)}.

(87)

Mn (x) = iRes S(k)ik=i~

(88)

Setting up the :first step o 2

we obtain N

o

Q (x;t) = -i.L:Res S(k)ik=ibn(x)exp[-bn(x)t] n

=L An(x) exp[-bn(x)t], N

(89)

n

where

A { ) _ 2bn(x)(bn(x) + Kn(x)) liN (bn(x) + K~(x))(bn(x) + b~(x)) n';tn (bn(x)- b~(x))(bn(x)- K~(x)) · (90) (bn(x)- Kn(x)) nX -

89

The corresponding formulae of the Bargmann approach [37] can be derived if we o put tcn(x) ll:n and bn(x) bn. Inserting the kernel Q (x;y,y') (89) into the parametric equations of the inverse problem (56)-(58) we obtain

=

=

d2

o

V (x;y) = -2 dy2 lndet IIP(x;y)ll;

(91)

o

f ± (x; k, y~ exp(±iky) ;... A ( )P-l ( . )exp[-(bn(x) + bn'(x) =f ik)y] X nn' x, Y (bn(x) =f ik)

(92 )

+~ n

where Pnn'(x; y) is defined as follows

P ,( . ) _ 8 , A ( )exp[-(bn(x) + bn'(x))y] nn X, Y - nn + n X bn(x) + bn'(x) · Parametric family of phase-equiva lent potentials At the second stage we find a family of potentials and solutions for normalizing constants that do not obey the condition (88): iRes S( k = itcn (x)) < M; < oo. To obtain an analog of the phase-equivale nt family of potentials for the parametric inverse problem, we should write the kernel of the integral Eq.(56) in the following form

M;

N

0

2

o

o

Q(x;y,y') = L(M~(x)- Mn (x)) f (itcn(x),y) f (itcn(x),y')

(93)

n

because the S-matrix is not changing now. And similarly, the kernel of the generalized shift KM (x; y, y') is written N

o2

o

KM(x;y,y') =- L(M~(x)- Mn (x))f(itcn(x),y) f (itcn(x),y').

(94)

n

Inserting KM(x;y,y') and F(x;y,y') into the basic parametric Marchenko Eqs.(56-58) we derive the following relations for the Jost potential and solutions o

V(x; y) =V (x; y)

d2

+ 2 dy 2 lndet P(x; y);

(95)

o

f±(x; k, y) =! ± (x; k, y)-

N

2o

L(M~- Mn) nm

0

f (itcn(x), y)P,;,;;(x; y)

1

00

Y

o

o

f (itcm(x), y') f ± (k, y')dy'.

90

The explicit dependence on fast variables is defined by the Jost solutions (92) determined at k = i~~:n(x), i.e. on the potential curves parametrically dependent on the slow dynamical variable x. Here we employed the notation

Pnm(x; y) = Cnm

o 2

+ (M~(x)- Mn

(x))

100 f (i~~:n(x), y') f (i~~:m(x), y')dy'. Y

o

o

So, we have considered certain examples of exactly solvable parametric models to show how the technique of Bargmann potentials is extended to the parametric family of the inverse problems. For a given functional dependence of spectral characteristics on the external coordinate variable, one may derive a large class of exactly solvable multidimensional models on the hasis of the parametric inverse problem for equations of a lower dimension. In such a way one may also investigate properties of a gauge equation system substituting obtained analytic functions of the hasis equation for a given analytical behavior from slow variables in induced gauge vector and scalar matrix potentials. This approach can be applied also to search for analytic solutions of nonlinear evolution equations. Thus, the parametric inverse problem is of primary interest by itself and not only as a constituent of the solution of the initial multidimensional problem in the adiabatic approach.

3 Supersymmetry of gauge equations and geometric nonadiabatic phases Conditions allowing supersymmetry for a system of gauge equations are special and strongly limited: the vector and scalar components of the gauge field ought to satisfy the principle of minimal coupling. That is why it is natural to want to widen opportunities of the approach by introducing an additional scalar potential but also conserving supersymmetry. Is this possible and will the additional scalar potential influence the geometrica! phases and topologica! effects? As it turns out, the additional scalar potential, conserving supersymmetry, affects the degeneracy of the ground state and may lead to its increase and vice versa to its decrease up to a vanisping of the degenaracy. Consequently, this leads to a vanishing of the conditiona for Wilczek and Zee phases and topologica! effects and to a spontaneous breaking of supersymmetry. An action of scalar potential is analogous to an action of the matrl.x field strength tensor FJJv· Nondiagonal elements of the induced connection operator A realise transitions between states of the parametric so-called instantaneous Hamiltonian hf and generate nonadiabatic Aharonov-Anandan phases (12, 48]. In the realistic statement of the three-body problem [3, 7], the nondiagonal matrix elements of A have to be taken into account since without them it is impossible to solve the problem correctly. Moreover near the level crossing, the adiabaticity is not valid. When the level crossing or quasicrossing between two or even three terms takes place, singularities of Anm arise and as a result, so do additional relevant geometric phases. This makes it necessary to introduce nonadiabatic AharonovAnandan matrices of geometrica! factors in the presence of A singularities.

91

The relationships hetween the gauge vector and scalar potentiala are obtained as a direct generalization of the Witten construction of SUSY [19]. Moreover, the interrelation hetween supersymmetric gauge equations and supersymmetric Schroedinger equations is stated. We also connect two directions of investigations in the gauge field theory: supersymmetry and geometric phases [52, 10]. 3.1 Supersymmetry

lnduced gauge potentiala appear naturally in the adiabatic approach to the description of quantum-mechanical systems dependent upon slowly varying externa! parameters and upon fast varying intrinsic ones. In this case the total Hamiltonian is decomposed into H = H 31 + H f, where H f (R) is the parametric family of the "fast" Hamiltonians depending on slow variables. The searched wave function of H is expressed as a sum over the eig~functions = bnm and completeness 1n >< n 1= 18(r-r') of the eigenstates I of the Hamiltonian H f at fixed R and get the "slow" system of equations of the gauge type (5) for the expansion coefficients Fn. Remember, F = {Fn} is a column-vector of dimension M and A, V are the vector- and scalar-potential-like matrix (6) Anm(R) =< (r)

= [detFot 1 detFo(tPo),

which is identica! to Eq. (42) of AB. With the matrix F2k(m) defined for k F(i,i)

_ {

2k(m) -

< -m by its elements

FJ~·j) f(i)

2k

(39)

(j

f.

(. _ J-

m)

)

m'

(40)

134

one obtains for the added bound states the wave functions

.,p(m)(r) = a:J 2 [det Fot 1 det Fo(m)·

(41)

They are equivalent to Eq. (43) of AB.

6 General spectrum modification For a fixed bound spectrum, finding the most general phase-equivalent potential is a well-understood problem [16] (see also AB for the supersymmetric approach). Therefore, I focus here on finding one potential whose energy spectrum has been modified in a given way. Other potentials with the same spectrum can then be derived with the standard technique in a further step. The principle of the calculation is very simple. First, the M states with labels so to BM-l are suppressed with the iterative procedure described in §2. This provides a potential V2M. Then, N states are introduced with the technique explained in §5. Eq. (37) leads to the potential (42) where F2M is a matrix whose dimension is N. The N states -1 to-N belong to the spectrum of the potential defined in (42), and the M states so to BM-l are suppressed. Now, V must be expressed as a function of Vo. To this end, let me introduce the auxiliary symmetric matricea X 2k with elements F.(i,i)

X(i,J) 2k

= f.r {

2k

00

f(i).,,(s;)dt 21; '1-'2k

-1/t.(i,JJ 2k

(-N < i J. < O) ' (-N -< i