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' The p r M e n c e of a ^ ^ thi duct shows that the in^ ^ 2 ^ £ ^ ^ J v b i c h could be introdescription of the electromagnetic field X ^ n t e r a c t s W l th the gravitational field Thus, in coS y ariant p e r ^ b a t i o n the ory for the electromagnetic and gravitational fields a fictitious neutral scalar particle participates in addition to the elements which have been described above. The reader who has understood me basic principles C 0 nstruct l 0 n of the diagram technique for gauge fields 4 approqpriate ^iculations J , '.r** *»f *• fortoemore comP il cated case. THE HAMILTONIAN FORMULATION OF RAVITATiniM THEORY r ^ A A justification of the correctness of the expression (3.13) for the invariant measure is based on the Hamiltonian formulation of gravitation theory. This formulation has been developed by D i r a c [ 1 4 ] . A series of variants for this formulation were obtained by diverse a u t h o r s ^ " 1 9 1 . The construction of an explicitly Hamiltonian form of the Einstein equations runs into the difficult problem of finding solutions to the constraint equations. For us it will be sufficient to consider a generalized Hamiltonian formulation of gravitation theory, . * . , \ . ' t where it is not necessary to solve the constraint equations, and one can restrict one's attention to a verifica tion of the commutation relations. We explain the generalized Hamiltonian formulation of the example of a system with a finite number of de^ * formulation the action funcf ^ ^ consideration ^ the form
in the action giving rise to vertices which are proportional to 6 (4) (0). The appearance of such renormalization terms is noted in many papers treating nonlinear theories (cf., e.g., C 4 0 ' 4 ^). We note that they are absent in the exponential parametrization. In this parametrization the measure (3.13) has, up to a constant factor, the simple form
Here p, q denote canonically conjugate coordinates and momenta which form a phase space of dimension 2n, Si is the Hamiltonian, <pa are the "constraints," A a are Lagrange multipliers (a = 1 m, m < n). The constraints <pa and the Hamiltonian & are in involution, i.e., satisfy the conditions
without any local additions. We have considered in detail the case of the gravitational field in vacuo. Introducing interactions with other fields does not change substantially the scheme of con struction of perturbation theory. For matter fields with
these ns usual Pois SO n brackets
SS^^i^S^^r^^sS^: s i 0 T i " a s fi\Tto,a
t i d e s and their correspondmg diagramfappear o n l / when a field with larger gauge group than the gravitational field is included, e.g., the electromagnetic field or fields of the Yang-Mills type. We shall not consider this case in detail here. We just list, as an example the expression of the functional integral corresponding to the electromagnetic and gravitational fields:
'
£°ed * » ~ £ £ * J ? " £ the m constraint equations
Tehductionof thebdimT £
£^t^rmine^t "
and the m supplementary conditions The functions x a are subject to the conditions
784
Sov. Phys.-Usp., Vol. 16, No. 6, May-June 1974
L. D. Faddeev and V. N. Popov
784
73 In addition it is convenient to assume that the funcThe variables r P „ which differ from the r ^ are not tions x a commute with one a n o t h e r : d y n a m i c a l variables. They can be excluded with the help of the constraint equations In this case it is simple to introduce canonical varia bles on the submanifold r * . Indeed, in view of (4.7), a canonical transformation in r allows one to go over to new variables, where the >•a take the form
The tem (4 14) contains the equations (4.10) together with the equations
where p a are part of the canonical momenta of the new system of variables. Let q a denote the coordinates conjugate to them and p*, q* the other canonical variables. In the new variables the condition (4.6) has the form { '
The solution of the system (4.10), (4.15) expressing the 'nondynamical" quantities r ° n , rk r k in terms of the l0 l0 1J n ,,„ T^. and h ^ " is of the form
and can be interpreted as the condition for solvability of the constraints <pa = 0 with respect to the coordinates q a . As a result of this the surface r * is defined in r by the equations
, where r . k are the three-dimensional connection coeffii]
so that the p* and q* play the roles of independent variables on r * . By construction these variables are canonical. A more detailed discussion of the generalized Hamiltonian formulation for the finite-dimensional case with applications to the theory of the Yang-Mills field is contained in a paper by one of the a u t h o r s ^ . Let us return to the gravitational field. We shall show that its action can be reduced to a form which is a fieldtheoretic analog of (4.1), with the appropriate constraints and the Hamiltonian satisfying conditions of the type (4.2). We shall follow the general method proposed by one of the authors in a form especially adapted for the gravitational field C 1 9 ] . For our purposes it is convenient to make use of a formalism of the first order. We consider the expression of the action for the gravitational field in the form (2.3) and collect in the corresponding Lagrange function all the t e r m s which involve derivatives with respect to
This expression does not contain the variables r £ which occur in y(h, r ) linearly and play the roles of Lagrange multipliers. The factors (denoted by A°°) in front of r £ are the constraints. The constraint equations
allow us to express the variables rL , (r?« - r%) in 0 l0 n HU , ^ terms of the r £ and h^ ". Then the t e r m s containing time-denvatives take the form
if one omits the terms
cients, defined by the three-dimensional metric g i k (i, k = l, 2, 3). us substitute ^ expression (4.16) for the rf 0 , k r k into the ^ g ^ e function x(h, V). After l 0 6 v MO'Mj * ' omittinS several terms of * e ^ of a divergence, which vanish when integrated over three-space when the asymptotic conditions (2.2) are taken into account, the result of the substitution reduces to the form
where
here g3 = d e t g i k , R3 is the three-dimensional curvature scalar associated to the three-dimensional metric g l k M k, = 1, 2, 3). The symbol v k in the expressions for the constraints Ti denotes the covariant derivative with respect to the metric g ^ . As pointed out by Arnowitt, Deser and Misnercis :l , the canonical variables and the expressions for the con straints have an intuitive geometric meaning. The functions qik and "ik serve as the coefficients of the first an °°, e -> 0, TVe = t" - t\ corresponds to the integration over an infinite number of variables we can formally present as p{t\ q(t), t' ^ is obtained simply by extendind the time in tegration to the whole axis. Let us rewrite the expression we should get in the limit using the source n(x,r) rather than its Fourier transform y(k,t),
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L.D. Faddeev
where Sn(a*,a) stands for the kernel of the S-matrix for scattering on the source T?(JC). It is clear now that it will be more elegant to use the normal sym bol. Indeed, to do this we are to drop the first term in the exponent. The re maining terms are manifestly Lorentz covariant. To see this let us introduce the solution of the free Klein-Gordon equation
and the Green function of this equation
It is easy to see that t, f->-°°and the outgoing wavea*(fc,f) ~a*(k)ei<jJt, t -*■ °°, The boundary terms take care of the subtractions necessary to make the action functional on such trajectories converge. The most natural perturbation scheme based on this definition is given by the stationary phase approximation to the functional integral. To realize it, it is convenient to restore the Planck constant h, which until now we considered to be equal to 1. The Feynman functional will acquire the form exp{(z'/fc X ac tion}. Of course ti may enter the action itself through the mass term, for ex ample. This "internal" fi will be held fixed in the following and we will pro ceed with the construction of the power series expansion only in the "exter nal"/!. The main contribution to the Feynman integral will be given by the value of the integrand in (4.5) on the critical trajectory. We already know that the boundary terms in (4.5) have such a form that the critical trajectory is given by the classical equations of motion:
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L.D. Faddeev
The solutions subject to the boundary conditions (4.6) have a limit for t" -*■ °° and t' -* —°°; the asymptotic form can be found by the iteration of the follow ing integral equation:
Introducing the field 0c[(x), where
we see that it satisfies the classical field equation
and that the integral equations (4.7) are equivalent to the integral equation
where Q(X) is given by (4.4) and is uniquely connected with a*(k) and a(k). The argument 0 C | on the r.h.s. is a solution Q through the integral equation (4.8). The tag "tree" on S means that the com puted approximation to the S-matrix contains the contribution to the normal coefficient functions of the S-matrix from the tree diagrams of ordinary per turbation theory. The proof of this is left as an exercise. It is instructive to observe that the exponent in (4.9) can be considered as an action functional
computed on the classical solution = 0C, in a specially prescribed regularized sense. Indeed we have formally
which is equal to the exponent of (4.9) if we agree to integrate by parts in the first two terms, use the free-motion equation to eliminate them and not care about the boundary terms. The next approximation is given by a Gaussian integration over the small deviations from the critical trajectory. Let us make the change of variables in the functional integral (4.5),
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L.D. Faddeev
We can now use as well the real variables
instead of the a and a* but the boundary conditions are more easily described in the former. The quadratic form in the deviations (*), 7r(x) is given by
where the particular choice of the first term on the r.h.s. is irrelevant due to the boundary conditions on 0 and if. The Gaussian functional integral
can be evaluated to give
where the operator ( - □ - m2) l is to be represented by the Feynman propa gator. The functional S0(0) generates the contribution to the normal symbol
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Functional methods
27
of the S-matrix from the diagrams with one closed loop. To check it is a sec ond exercise. It is clear now how to proceed to the higher orders in (ft) 1 / 2 . One decom poses the action written in the variables $ and if,
and performs the Gaussian integration. There appears the natural generaliza tion of the Feynman rules. The quadratic form (1/ft) ^? 2 supplies the propa gator which is a Green function = Wl(R.n), define self-adjoint o p e r a t o r s , which we will denote by operator H0 has absolutely continuous s p e c t r u m . Its diagonal representation is r e the Fourier transformation
Here
The assertion that the operator H has the same absolutely continuous spectrum as H0 is the fundamental r e s u l t of scattering theory. More precisely, there exists an invariant decomposition relative to H of the space J? in the direct orthogonal sum Translated from Itogi Naukii Tekhniki. Sovremennye Problemy Matematiki, Vol. 3, pp. 93-180, 1974. ©1976 Plenum Publishing Corporation. 227 West 17 th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.
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of natural subspaces corresponding to the discrete and absolutely continuous spectrum of this opera t o r . Here the restriction of H to £ a c . is unitarily equivalent to H 0 . There exist among the o p e r a t o r s i s o m e t r i c in £ that realize this equivalence two o p e r a t o r s IP*' d i s tinguished in t e r m s of their physical origin. They are called wave o p e r a t o r s and a r e defined by
where the limit is understood in the sense of a strong operator topology. There exists a broad l i t e r a t u r e that deals with the conditions on v(x) under which these limits exist. Our problem does not include the presentation of this "direct" problem of scattering theory, although s e v e r a l r e s u l t s relative to the e x i s tence of operators t r * ' will also be mentioned in the text. Details on this question may be found, for ex ample, in the monograph of Kato [36], We note that the fact that the operator V = H - H0 is a function-mul tiplication operator plays no role in general discussions in scattering theory. The operators TjW a r e i s o m e t r i c :
Here P is a projector on the subspace £<j. which, as a r u l e , is finite-dimensional. The second relation is said to be a completeness condition. The unitary equivalence spoken of above is realized by the equation
The physical meaning of wave operators is based on the following concepts. In quantum mechanics the operator
d e s c r i b e s the evolution of a system, which in our c a s e , consists of a particle in the field of a potential center. Over a long period of time a particle with positive energy exits far from the center, becoming non-sensitive to its influence, and as a result its development over the course of time as |r|-*oo is actually described by the operator
corresponding to free motion. More precisely we may associate with every single-parameter family of vectors l^-(0> describing free motion (wave packet)
a solution of the Schrbedinger equation
such that
as t->- — oo. The precise equation defining such a solution and following from the existence of wave o p e r a t o r s , has the form
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123 Every solution of the Schrdedinger equation tf(t) from the given class as t — °° again reduces to a wave packet, in general differing from ^_(t):
as t — °°, where
The l a s t equation is justified since the operators U and u(~) have a common range of values. The passage from the wave packet i^_(t) describing the initial state of a particle, to the wave packet 4+(t) describing its final state is also the p r o c e s s by which a particle is scattered by the center. All in formation on this p r o c e s s is contained in the operator S, which r e l a t e s both wave packets by the formula
Comparing the equations expressing 4 in t e r m s of 4- and 4 + in t e r m s of 4, we see that
where it follows from the properties of Tj(+) and u( ' that S is unitary and commutes with H0.
These relations reflect conservation of particle and energy flux in the course of scattering. The operator S is said to be the scattering operator. Its representation
in a diagonal realization of H0 is defined by the equation
where the 5-function in the integrand explicitly takes into account the fact that S and H 0 commute. The function f(k, l) defined for | k | = 11\ is called the scattering amplitude. T h e r e exists an alternative approach to the scattering theory, called the stationary approach, based on the study of the asymptotics of the eigenfunctions of H as |x| — » . The scattering amplitude f(k, I) h e r e explicitly occurs in the description of these asymptotics. Such an approach and its relation to the nonstationary approach will be illustrated in the text. The problem of reconstructing the potential v(x) corresponding to the scattering amplitude f(k, /) is said to be the i n v e r s e problem of scattering theory. This problem is not defined if the perturbation V = H - H0 is an a r b i t r a r y operator, since an entire set of operators V can easily be selected with respect to an a r b i t r a r y unitary operator of the form S, such that the corresponding operator S is a scattering operator for the pair H0 and Ho + V. It becomes meaningful only under a further condition, that V is an operator of multiplication by a function. Henceforth, this condition will be said to be the locality of the potential. Over the past 25 years the inverse problem has been solved for the case most interesting for physical applications of a spherically symmetric potential, viz., when n = 3, and
In this case the scattering amplitude f(k, I) depends only on the lengths of the vector k and I, which a r e equal by the condition, and on the angle between them so that it is actually a function f(|k|, cosfl) of two v a r i a b l e s . The partial scattering amplitudes fj(|k|) a r i s e in decomposing f(|k|, cos6) in Legendre poly nomials
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124 so c h a r a c t e r i s t i c for spherically s y m m e t r i c problems. The unitarity condition on an operator S can be explicitly borne in mind by setting
where i)j(|k|) is a r e a l function, called the asymptotic phase because of its role in the stationary formula tion of the scattering problem. The fundamental result obtained by V. A. Marchenko [16] and M. G. Krein [12] states that the potential v(x) is reconstructed by one of the asymptotic phases TJ,( | k | ) , one such phase being an arbitrary real function satisfying the integrality condition
Here m positive numbers associated with the characteristics of a discrete spectrum of the radial Schrodinger operator
whose reconstruction constitutes our problem, must be specified for a unique determination of the potential by a given phase rj,. It is precisely this result that was dealt with in the survey [25]. The subsequent development of the inverse problem, about which we shall speak in the current article, is associated with the Schrbedinger operator in the general case without spherical symmetry type assumptions. Here, two cases a r e distin guished, theoretically differing in t e r m s of technical difficulties: n = 1 and n S: 2. In the first c a s e , tools developed for the radial Schrbedinger operator with 1 = 0 turned out to be applicable. The chief role is played by the existence of a fundamental system of solutions for the corresponding one-dimensional dif ferential equation. In the multidimensional case, when we must deal with a partial differential equation contrary to ordinary differential equations, the concepts of a fundamental system vanishes. It may be that it is precisely this circumstance that constituted the hindrance that has extended the study of the multidimensional inverse problem over such a long period of time. In spite of this circumstance, which constitutes a technical distinction, it turned out that the one-di mensional and multidimensional inverse problems are to some degree analogous in many ways. This analogy can be particularly seen in an operator-theoretic language, which we have chosen for our p r e s e n t a tion for precisely this reason. It is of interest that all these assertions on analogy refer to the one-di mensional, but not radial Schrbedinger operator. In this sense the one-dimensional case plays a fortunate role as an intermediate link between the radial Schrbedinger operator and the multidimensional operator, being technically close to the former and conceptually anticipatory of the fundamental outlines of the l a t t e r . We will now indicate the principal distinction between the inverse problem considered in this survey and the case of the radial Schrodinger operator. It consists in the overdeterminacy of this problem. In the radial case we must construct a function v(r) decreasing at infinity of a variable r that varies on the half-axis, in t e r m s of a function r\i(\k\) of a variable |k| also on the half-axis and also satisfying an a s y m p totic condition as |^| —-co. it is therefore not remarkable that the function r\t(\k\) can be chosen a r b i t r a r i l y . A similar simple calculation of p a r a m e t e r s demonstrates that the scattering amplitude for m o r e complex problems cannot be arbitrarily selected. Let us first consider the Schrbedinger operator for n = 1. An a r b i t r a r y unitary scattering amplitude f(k, 1), where k, I € R1 can be parametricized by four r e a l functions of the variable |k|, running through the half-axis. A symmetry condition, which follows from the r e a l n e s s of the potential and which will be p r e sented in the text, d e c r e a s e s this number down to three. At the same time the potential v(x) can be con sidered only as two real functions defined on the half-axis. Indeterminacy is present since it is difficult to imagine the problem of the physical origin in which the nondegenerate correspondence of sets of two and three arbitrary functions would be established. In other words these heuristic arguments demonstrate that the scattering amplitude f(k, /) will satisfy the additional necessary condition and so can be expressed in terms of two real functions of the half-axis. Such a condition in fact arises and is derived in the text.
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125 When n a 2 this indeterminacy is significantly aggravated. The scattering amplitude here is a func tion of 2n— 1 variables, while the property of being unitary reduces to this function being r e a l . At the s a m e time the potential is a r e a l function of n variables. As a r e s u l t of such indeterminacy the problem of determining necessary conditions on the scattering amplitude that follow from the locality condition on the potential, arises and becomes of great importance. It was previously unclear that such conditions can in general be expressed in terms of the scattering amplitude in sufficiently explicit form. Nevertheless, as will be explained in the text, such conditions can be described. We will now formulate the fundamental statements of the formalism of the inverse problem. We will a s s u m e for the sake of definiteness that the operator H has one simple eigenvalue, so that the projector P under an isometricity condition is a projector on the one-dimensional subspace the corresponding eigen vector u spans. The choice of the transformation operator U, i.e., the choice of the solution of the equa tion
which differs from wave o p e r a t o r s , is the basis of this approach. Every transformation operator is ob tained from the wave o p e r a t o r s u W by multiplication on the right by a normalizing operator factor N^"' that commutes with H0,
In p a r t i c u l a r the scattering operator S is a normalizing factor for U^"^ with respect
Comparing these two formulas we can see that a factorization of the scattering operator
c o r r r e s p o n d s to every choice of the transformation operator U. We now assume that U is invertible in the sense that there exists a vector \ not belonging to the space j? , such that
Then the completeness condition expressed in t e r m s of U will be
where
The operator W will be called a weight operator. Equations (1) and (2) constitute the basis for solving the inverse problem. We must find a s u c c e s s ful determination of U, such that Eq. (2) uniquely determines it in t e r m s of the weight operator W and that the corresponding factors N W is uniquely determined by the factorization condition of Eq. (1) in t e r m s of a given operator S. It turns out that such transformation operators exist and are distinguished by a Volterra property. Let us clarify in detail what we understand by a Volterra property. In the one-dimensional case this concept is formulated in the most classical fashion. The kernel A(x, y), where x, y € R ' is said to be t r i a n gular if A(x, y) = 0 when x < y or A(x, y) = 0 when x >y. An integral operator with triangular kernel is said to be a triangular operator. Finally an operator of the form "identity element plus triangular operator" is said to be a Volterra operator. We have two possibilities for a Volterra operator in the one-dimensional case:
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Both formulas may be described uniquely if a direction y is introduced, i.e., actually a variable taking two values y = ± 1
In this form the calculation of the Volterra operator is naturally carried over to the case n S 2 . The variable y in this case is a unit vector and runs through, unlike the one-dimensional case, a connected set, namely the sphere S n _ 1 . An operator of the form
is said to be a Volterra operator with direction y of Volterra property. We will now prove that the Volterra property of a transformation operator Uy reduces the complete ness equation (2) to a linear integral equation for the kernel Ay(x, y) occurring in its definition. Suppose y is some direction and let Uy be a Volterra transformation operator with direction y of Volterra p r o p e r ty. We consider the operator
This operator has direction of Volterra property opposite to that of the operator y , so that
Suppose W
is a weight operator for Uy, and we set
Equation (2) with the notation introduced, can be rewritten in the form
or, in m o r e detail, in t e r m s of the kernels A (x, y), fty(x, y), and Ay(x, y) in the form
The right side h e r e vanishes when (y - x, y)> 0. If this condition holds, we obtain the linear integral equa tion
which can be used to find the kernel A (x, y) in t e r m s of the known kernel S7y(x, y). This equation consti tutes a general formulation of the Gel'fand-Levitan equation introduced in [8] for the actual example of a S t u r m - L i o u v i l l e operator on the half-axis. Thus we will see how to reconstruct the Volterra transformation operator U y if the corresponding weight operator W y is known. To construct the operator W y in t e r m s of a known operator S it is n e c e s sary to solve one m o r e problem in the factorization of Eq. (1) to determine the normalizing factors of U y .
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It is still not evident whether this problem for Volterra U r a l s o reduces to a linear equation nor whether it is even solved explicitly in the one-dimensional case. However it turns out that normalizing factors for Volterra transformation operators themselves turn out in some sense to be Volterra. We will not clarify this circumstance in more detail here and refer the reader to precise formulations in the text. Thus, the procedure for solving the inverse problem for given scattering amplitude is reduced to a set of factorization problems (1) to determine the normalizing factors N ^ . A weight operator W y is constructed in terms of these data and characteristics of the discrete spectrum, if such exists, and then using the Gel'fand-Levitan equation, the transformation operators U y a r e reconstructed. All the stages, in general, can be realized for an arbitrary initial operator S. Moreover if S is unitary, each of the operators
will be self-adjoint. Additional necessary conditions, about which we spoke above, begin to play a role at the next stage, when it is clarified that the operators Hy in fact are independent of y . This important s t a t e ment simultaneously s e r v e s for investigating the properties for the reconstructed operator H and for prov ing that the initial operator S is in fact the scattering operator for the pair H and H0. The tools for proving the independence of Hy from y differ for n = 1 and n a 2, because of the difference between the range of values of the variable y . When n ^ 2 , i.e., when this set is connected, we can use differentiation with r e spect to the p a r a m e t e r y . In the one-dimensional case it is necessary to use more artificial m e a n s . We conclude this description of the tools for solving the inverse problem, since further detail r e quires a m o r e formal presentation, which will be presented in the text. We note only that, in our opinion, abstract scattering theory can be further developed so that Volterra transformation operators and the existence of separated factorizations of the form (1) of the scattering operator find a natural place within its framework. Apparently the formulation of scattering theory due to Lax and Phillips [45] is the most successful starting point for such a generalization, and a suitable causality condition will in a reasonable way be the appropriate language. Let us now indicate on the structure of the survey. Differences in the technique and elaboration of the cases n = 1 and n > 2 forced us to treat them separately. We will emphasize the analogies between the corresponding discussions and equations wherever possible. The one-dimensional case is discussed in Chap. 1. A significant technical simplification for study ing the one-dimensional Schrodinger operator lies in the existence of special fundamental systems of solutions of the correspondinding differential equation. All operator equations a r e suitably introduced and justified proceeding on the b a s i s of the well-known properties of these solutions. The description of these p r o p e r t i e s is discussed in Sec. 1, which plays an auxiliary role. In Sec. 2 the fundamental statements of scattering theory for a given concrete example are formulated and proved. Volterra transformation opera t o r s a r e introduced in Sec. 3 and the normalizing factors corresponding to them are obtained in Sec. 4. G e l ' f a n d - L e v i t a n - t y p e equations are formulated in the latter section. Section 5 t r e a t s the solvability of these equation. A relation is analyzed there between transformation o p e r a t o r s for y = 1 and y = — 1 . The general investigation of the inverse problem concludes h e r e . The last Sec. 6 contains a description of an explicit solution of the Gel'fand-Levitan equation for the particular case when the scattering amplitude is a rational function of a p a r a m e t e r k. Chapter 2 also t r e a t s one-dimensional problems. Here a generalization of the formalism developed in Chap. 1 to the case of potentials v(x) having nonzero asymptotic a s x » (Sec. 1) or to the case of an o p e r a t o r of the form
which is a d i r e c t generalization of the Schroedinger operator (Sec. 2), is analyzed at an elementary level. In the l a s t section we will describe so-called t r a c e identities, which relate certain functionals of the poten tial and scattering amplitude. These identities, through not a means for the direct solution of the inverse problem, can indirectly lead to information on the potential according to known properties of the s c a t t e r ing amplitude, and conversely. Section 4 d e s c r i b e s an application of the inverse problem of scattering theory to the solution of one-dimensional nonlinear evolutionary equations. The starting point of this application was set forth in the important work of Kruskal et al. [42], Then P. Lax [44], V. E. Zakharov and A. B. Shabat [11], V. E. Zakharov and the author [10], and others further developed this subject. The inverse
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problem method of scattering theory for solving nonlinear equations currently draws ever g r e a t e r attention and is rapidly developing. The region of its applicability is still far from clarified and we will consider in the present survey only two c h a r a c t e r i s t i c examples. In Chap. 3 we r e t u r n to our fundamental theme and consider the inverse problem for the multidimen sional Schrbedinger operator. For the sake of definiteness we will deal with the physically interesting case n = 3, although all the discussions are trivially carried over to a r b i t r a r y n s 2. In the multidimen sional case no fundamental system of solutions nor proof procedure is available due to the far g r e a t e r c u m b e r s o m e n e s s . The scope of the present survey does not allow us to present somewhat instructive estimates needed to make all the constructions of Chap. 3 r i g o r o u s . We will therefore limit ourselves to a presentation of only the formal scheme of these constructions. We leave it to the r e a d e r to complete the algebraic framework of this scheme by appropriate analytic arguments. In Sec. 1 we set forth the fundamentals of scattering theory for the three-dimensional Schrodinger operator. General concepts that are of assistance in research on Volterra transformation operators U r a r e described in Sec. 2. The construction of normalizing factors and the weight operator W y a r e discussed in Sees. 3 and 4. The description of the Gel'fand-Levitan operator and a scheme for studying the inverse problem are presented in Sec. 5, with which Chap. 3 concludes. No special knowledge is required to read this survey. In particular, it can be read independently of the preceding survey [25], since all the necessary information on the Schrodinger operator a r e enumerated h e r e once again. We hope that for some mathematicians this survey can s e r v e as an introduction to s c a t tering theory, a branch of functional analysis and mathematical physics which is constantly expanding the domain of its applications. In concluding this introduction we note trends associated with the inverse problem of scattering the ory that a r e not indicated in this survey. These include: 1. works of B. M. Levitan and M. G. Gasymov, M. G. Krein, and his students on canonical systems and Dirac-type s y s t e m s on the s e m i - a x i s . These works in t e r m s of the formulation of the problem and methods relate to the problems associated with the radial Schrbedinger operator. We refer the r e a d e r to new studies [7, 13] for references to the l i t e r a t u r e given there. 2. Works on inverse problems in t e r m s of scattering data for fixed energy. By this problem is under stood the reconstruction of the potential v(r) in t e r m s of a known set of asymptotic phases r?;(|k|) for all I = 0, 1, 2 , . . . and fixed |k|. An operator-theoretic formulation of this problem is not at all evident and the r e s u l t s obtained have yet to reach, in t e r m s of elegance and completeness, the level attained in the s p e c t r a l formulation of the inverse problem. The most detailed presentation of well-known facts on this p r o b lem can be found in Loeffel [47]. 3. Works of V. A. Marchenko and his students on the stability of the inverse problem, p r i m a r i l y for the example of the radial Schrbedinger equation. The recent monograph of V. A. Marchenko [17] d i s c u s s e s this subject. We will use no unusual notation. Constants appearing in the limits a r e denoted by C. An explicit d e pendence of these constants on p a r a m e t e r s is indicated only if this is important. In numbered equations the first digit indicates the number of the section and the second, the number of the equation. The number ing of the sections begins anew within each chapter. A reference to an equation of a different chapter will use a number made of three digits of the type (II.3.14), whose meaning is self-evident. CHAPTER ONE-DIMENSIONAL
1
SCHROEDINGER
OPERATOR
In this chapter we will consider the Schrbedinger operator
where the potential v(x) is assumed to be a r e a l measureable function satisfying the condition
(P)
341
129 Here the o p e r a t o r H defined on the dense set £> = U^(R) in the Hilbert space i} = L2{R) is a self-adjoint o p e r a t o r . We will introduce and c h a r a c t e r i z e the scattering data corresponding to this operator and d e s c r i b e the procedure for solving the inverse problem for reconstructing the potential v(x) in t e r m s of these data. 1. F u n d a m e n t a l
System
of S o l u t i o n s
of S c h r d e d i n g e r
Equation
This section is auxiliary. Here we will describe two fundamental systems of solutions of the Schrde dinger equation
Henceforth k, as a r u l e , will be a real number, but sometimes we will a s s u m e it to be a complex number, particularly when specified. Condition (P) means that v(x) effectively vanishes as |JC|-> oo, so that we may naturally assume that every solution of Eq. (1.1) coincides at infinity with some solution of the equation
i.e., a linear combination of exponents
More rigorously, we may prove that there exist solutions f ( (x, k) and f2(x, k) of Eq. (1.1) which have the asymptotic
The proof is based on the fact that the differential equation (1.1) with the boundary conditions (1.2) and (1.3) is equivalent to the equations
where
and 0(x) is the Heaviside function
These equations a r e Volterra-type integral equations, so that the method of successive approxima tions always converges for them. Here the p a r a m e t e r k can have complex values from the upper halfplane. As a r e s u l t of analyzing the successive approximations we will prove that the solutions f,(x, k) and f2(x, k) exist and for fixed x are analytic functions of k when 1m k> 0 and are continuous when Im k = 0. Here we have the bounds for them
342
130
Such assertions were first obtained by Levinson [46], It follows from the bound of Eq. (1.6) on the basis of the Jordan lemma that we have for the solution f^x, k) the integral representation
where the kernel A,(x, y) is quadratically integrable with respect to y for any fixed x. Similarly f2(x, k) can be represented in the form
where the kernel A2(x, y) also is quadratically integrable with respect to y. Such integral representations for solving the Schroedinger equation were introduced by B. Ya. Levin [15]. The detailed properties of the kernels Aj(x, y) can be obtained on the b a s i s of an investigation into integral equations equivalent to the corresponding equations (1.4) and (1.5) for the solutions f,(x, k) and f2(x, k)
Such equations were first derived by Z. S. Agranovich and V. A. Marchenko [1]. Agranovich and Marchenko proved the convergence of the method of successive approximations for these equations and obtained bounds on the solutions. To write the bounds it is suitable to introduce the monotone functions
Bounds on the kernels A^x, y) and A2(x, y) have the form
Further it is possible to prove using these equations the existence of the first derivatives of A^x, y) and A 2 (x, y) and to obtain bounds on them. For example,
and similar bounds hold for ^- A, (x, y) and ^-A2(x, y). Finally, it is evident from the equations that
343
131 so that
The p a i r s Ji(x,k), ft(x, — k)=fx(x,k) and / 2 (x, k), f2(x, —k) = /2(x, k) for real k * 0 are fundamen tal s y s t e m s of solutions of the fundamental equation (1.1). In fact, since the Wronskian {/,, / , } = / ! / , _ / , / , ' is independent of x, it coincides with its values as x—- °°, which may be calculated using the asymptotic for the solution fj(x, k) and its derivative. It can be proved that as x — °°
so that
We will see that when k * 0, the Wronskian is nonzero and the solutions fj(x, k) and f^x, - k ) are linearly independent. Similarly
so that f2(x, k) and f2(x,—k) a r e also linearly independent when k * 0. Any solution of Eq. (1.1) can be represented in the form of a linear combination of the solutions fj(x, k) and fj(x, - k ) or f2(x, k) and f2(x, —k). In particular, we have
Substituting Eq. (1.15) for f2(x, k) in Eq. (1.16) and performing the s a m e operation with fj(x, k) we find that the following equations must h o l d i n E q s . (1.15) and (1.16) a r e to be consistent:
We may e x p r e s s the coefficients Cjj(k), i, j = 1, 2 in t e r m s of the Wronskians of the solutions f,(x, k) and f2(x, k). In view of Eqs. (1.13) and (1.14) and also in view of the self-evident equations
we find
Comparing Eqs. (1.19) and (1.20) we find that
which, incidentally also follows from Eq. (1.17) since Cj2(k) -- (^(k). These equations imply also that
344
132 We will see that the four coefficients cij(k) a r e in fact expressed in t e r m s of two complex-valued functions
satisfying the condition
Here
We will henceforth refer to these functions as transition coefficients. To derive further properties of the functions c(k) and b(k) we e x p r e s s them in t e r m s of the k e r n e l A 2 (x, y). F o r this purpose we note that Eq. (1.5) implies that f2(x, k) a s x - « has the asymptotic
Comparing this equation with the equation
which follows from Eq. (1.15) if we take into account definition (1.2) of the solutions fj(x, k) we obtain for a(k) and b(k) the equations
We now replace f3(x, k) by the kernel A2(x, y) using Eq. (1.9). We find
where
and
where
345
133 Bounds on fl^x) and II2(x) follow from the bounds on A 2 (x, y):
which imply that the function n 2 ( x ) is absolutely integrable on the semi-axis 0 < x < °°, while fl|(x) is absolutely integrable on the entire axis. Thus, we have found for the transition coefficients a(k) and b(k) an expression in the form of a Fourier transform of absolutely integrable functions. In particular, it follows from the obtained representations that for large k we have for these coefficients the asymptotic
Moreover, we see that a(k) is the limiting value on the r e a l axis of a function analytic and bounded in the half-plane Im k> 0 and that the asymptotic of Eqs. (1.24) holds for all k with Im k ^ 0. We will consider the distribution of the zeroes of a(k) on the complex plane. Because of Eqs. (1.21), a(k) does not vanish on the real axis. F u r t h e r , it follows from the asymptotic of Eq. (1.24) that a(k) is also nonzero for sufficiently large |k|. It therefore follows that a(k) can have only a finite number of z e r o e s . It follows from the representation of Eq. (1.18) for a(k) by means of the Wronskian of the solu tions fj(x, k) and f2(x, k) that if a(k 0 ) = 0, these solutions are linearly dependent for k = k 0 , i.e.,
We note that when Imfc > 0 the solution fj(x, k) exponentially d e c r e a s e s a s x - » while the solution f2(x, k) behaves likewise as x ~ °°. When k = k 6 we may conclude on the basis of Eq. (1.25) that Eq. (1.1) has a solution that is quadratically integrable on the entire axis. The formal self-conjugacy of the equation im plies that this is possible only for real kjj, i.e., for purely imaginary k 0 . We have thus found that a(k) can have only a finite number of purely imaginary z e r o e s . We will prove that these z e r o e s are simple. For this purpose we obtain an expression for a(k0) = (d/dk)a(k)| k=k0We will proceed on the b a s i s of Eq. (1.1) for f t (x, k) and f2(x, k) and the equation
for fj(x, k) and f2(x, k). We obtain by the standard method the identities
On the other hand, using Eq. (1.18) we find
Suppose now that k coincides with one of the zeroes of a(k), which we again denote by k 0 . When k = k0 the Wronskians in Eqs. (1.26) taken for x = ± A vanish and the integrals in the right sides of these equations converge in limit as A —■«>. Comparing Eqs. (1.26) and (1.27) and recalling that o(k0) = 0, we find
346
134 The solutions f^x, k) and f2(x, k) for imaginary k are r e a l . The integral in the right side of Eq. (1.28) does not vanish because of Eq. (1.25), so that o(k0) * 0 and, consequently the zeroes of a(k) are simple. These z e r o e s will henceforth be denoted by in? .where 1 = 1,..., N. With this we conclude the study of the p r o p erties of the transition coefficients a(k) and b(k). In concluding this section we present an expression for G r e e n ' s function of Eq. (1.1). Suppose X is a complex p a r a m e t e r and let us select a branch for y T , such that I m ] / ) 7 > 0 . The kernel
for fixed x and y is an analytic function of X on the plane with section on the positive part of the r e a l axis and with simple poles at the points X = _ x - ; - If \ does not coincide with these points and if X * 0, we have this kernel the bounds
Here R(x, y; X), as follows from Eq. (1.18), is a solution of the equation
We may prove using these facts that the integral operator R(X) with kernel R(x, y; X) is the resolvent ( H ~ XI)" 1 of the self-adjoint operator H. Moreover, we may use the properties of this kernel to define the operator H itself, which incidentally we have not done. Let us now turn our attention to the fact that the function a (V^) is in the denominator of the resolvent R(x, y; X) and has zeroes at the eigenvalues of H. In this it reminds us of the c h a r a c t e r i s t i c determinant det(H - XI). We may verify that such an interpretation of it is in fact justified. F o r example, we have the equation
where R0(X) is the resolvent of the operator H0. Subtraction by R0(X) plays the role of a r e q u i r e d regularization for the definition of det(H 0 everywhere. The table of the asymptotics of these solutions as |x| — » has the form
The left column here being referred to x-+ — oo, and the right column to x-*- oo. The coefficients sjj(k) and iij(k) occurring in this table are expressed in the following way in terms of the transition coefficients a(k) and b(k):
These properties follow from relationships of the form of Eqs. (2.15) and (1.16) and the asymptotics of Eqs. (1.2) and (1.3) for the solutions fj(x, k) and f2(x, k). The functions Sjjfk) and s12(k) have meaning for all k * 0, since a(k) does not vanish on the real axis. We will prove that if |a(0)|= oo, the coefficients of s,, and s,2 equally have meaning up to k = 0. In this case, evidently, Su(0) = 0 and only s12(k) is to be considered. It is evident from Eq. (1.22) that |a(A)|-> oo, if
Here, as is evident from Eq. (1.23),
so that s12(k) is defined by continuity up to k = 0 and s12(0) = - 1 . We may similarly prove that in this case s21(0) = - 1 and s22(0) = 0. For larger |k|,
348
136 We can naturally continue Sjj(k) to the s e m i - a x i s k < 0 by the equation
We may easily verify based on the property of Eq. (1.21) that the m a t r i c e s
are inverses of each other, as is indicated by the notation, and are unitary matrices, so that for example
or, in more detail,
We will s e e , in particular, that for all k * 0,
and we recall that if |s 12 (0)| = 1, then
Finally, a comparison of the asymptotics of the set of solutions u> '(x, k) and uj '(x, k) implies the linear relation
where natural vector notation is used.
•
We note that the set of solutions uj (x. k) are naturally interpreted in their asymptotic in t e r m s of a radiation condition. However, we will not use this fact anywhere below. The solutions Uj (x, k) constitute a complete orthonormalized set of eigenfunctions of the continuous spectrum of the operator H. We may verify this fact by calculating the jump in the resolvent R(x, y; X) through a section in the positive part of the real axis, which corresponds to the continuous spectrum of H. We have the equation
which quite simply verifies the direct substitution of Eqs. (2.1) and (2.2) for the solutions UJ±»(JC, k) and u^±}(x, k) in t e r m s of f,(x, k) and f2(x, k) in the right side. The completeness equation which thereby fol lows has the form
Here the u/(x), I = 1 , . . . , N are orthonormalized eigenfunctions of the discontinuous spectrum of H. The orthogonality relation
349
137
can be derived using the identity
which is t r u e for a r b i t r a r y solutions of the Schrbedinger Eq. (1.1), the asymptotic as jc[—*■ oo of the solu tions U J ^ J C , k), and the unitary condition on S(k). We construct using the solutions u^ix,
k) two maps T±:S}-+fy0 using the equations
The completeness and orthogonality relations are written in t e r m s of these operators as follows:
Here P is a projector into £ on the proper subspace of H spanned by its eigenvectors u;, I = 1
N.
We will now prove that the wave vectors can be introduced by the equations
F o r the proof it is sufficient to demonstrate that for any vector - ± oo. This is in turn easy to verify. In fact, recalling the definition of the operators T ± and T 0 , we can write the functions x^'C*. t) representing the vectors x ( ± ) ( 0 . i n t n e form
The functions 2(k2) can be considered as continuous and finite. By the R i e m a n n - L e b e s g u e l e m ma the contribution to the integral
from an a r b i t r a r y finite interval |jc|tj, 7 = 1
N.
3) the same number of positive numbers m y ' , I = 1 , . . . , N. We construct using these data the func tion J2](x) in t e r m s of Eq. (3.14) and consider the equation
as the equation for A](x, y). This is an equation in t e r m s of the second independent variable of this func tion, where x occurs only as a p a r a m e t e r . Setting
we r e w r i t e Eq. (4.1) in the form of the operator equation
The free term w x (y) is absolutely integrable and bounded, and, consequently, is also a quadratically integrable function on the interval xor(y)g£,>(jc, oo). We will find a solution also from L 12 (x, ) and prove that it exists and is unique for any x, — oo < x < oo. For this purpose we first verify that we are dealing with an equation possessing a completely con tinuous operator in Lj(x, °°) and L 2 (x, °°). Suppose
358
146 The function TJ^X) on the b a s i s of Eq. (4.12) is absolutely integrable on the intervals [a, «=) for any a > — oo and we have the inequalities
Using the bound
we find that
i.e., the operator fix is a Hilbert-Schmidt type operator and its norm approaches zero a s x - « . The com plete continuousness of Qx in Li(x, °°) is also a well-known corollary of the absolute integrability of T)I(X). We now note that the operator I + Qx is positively defined for any x. In fact it is obtained by the restriction of the positive operator W from Sec. 3 to L 2 (x, °°). In particular, this implies that the homogeneous equation
has no nontrivial solutions in L 2 (x, ■»). We now prove that every solution of Eq. (5.2) in L,(x, °°) also b e longs to L 2 (x, °°). We have
so that h[(y) is quadratically integrable on the interval [x, °°). We conclude that Eq. (5.2) has no nontrivial solutions in L t (x, « ) , so that Eq. (5.1) is uniquely solvable in Lj(x, °°) for any x. We will now consider how the bounds for the solution A t (x, y) follow from this fact. The operator (I + fix)-1 is uniformly bounded for all x from the interval [a, ), since the norm of fix approaches zero as x — •*>
so that
Substituting this bound in the integral occurring in Eq. (4.1), we find that
Using Eq. (4.1) we may also verify that the solution A,(x, y) is once differentiable, and we may find bounds on the derivatives. Let us estimate the function -^Ax(x,y). Differentiating Eq. (4.1) with respect to x, we a r r i v e at the equation for bx(y) = ^— A\(x, y) + ^— 2,(x + y) of
Here the free t e r m
359
147 has the bound
We therefore find for the solution b^y) that
We may similarly estimate the derivative j - At(x, y) and the result is given by
The integral equation (4.2) is similarly investigated. If
This equation is uniquely solvable and we have for the solution the bounds
where
If the functions F,(x) and F2(x) have more than one derivative, the kernels A,(x, y) and A2(x, y) are also multiply differentiable. Bounds on the corresponding derivatives are found in a way similar to what was done for b ^ y ) . Returning to the pair of analytic functions u(x, k) and f(x, k), we verify that if we are given a func tion s(k) satisfying the conditions repeatedly formulated for s12(k) and N unequal positive numbers x/, I = 1 , . . . , N, and further N positive numbers m^, there exists a unique pair of functions u(x, k) and f(x, k), such that 1) the function f(x, k) and u(x, k) are analytically continued in the upper half-plane l i n k s 0, where f(x, k)e~il°c is bounded for all k, Im k s 0, and u(x, k) has simple poles at the given points k = ix/, 1 * 1 , . . . . N; 2) the residues u(x, k) are connected to the values of f(x, k) when k = ixj by the equation
3) on the real axis by
4) for large |k| by
5) for real k we have the equation
360
148 We now construct using the solutions for the Gel'fand-Levitan equations A,(x, y) and A 2 (x, y) found, operators Uj and U2 using Eqs. (3.1) and (3.2) and consider the operators
We c a r r y out the investigation of these operators at the formal level, without going too deeply into justifica tions. A rigorous justification of the results obtained here can be derived m o r e simply by following the quite elementary, but laborious discussions of [1], We first prove that the operators U\ are self-adjoint. For the proof we note that the G e l ' f a n d - L e v i tan equations derived from the completeness equation (3.13) a r e in fact equivalent to it, i.e., in other words the operators U( and U2 obtained by us satisfy this equation. We consider for the sake of definiteness the case i = 1. Suppose an operator A, has kernel A,(x, y), where
We construct the Volterra operator
The Gel'fand-Levitan equation can now be written in the form
The operator
is also self-adjoint since Wj is self-adjoint. But it is simultaneously a Volterra operator, since the opera t o r s Ui and Uj are Volterra with identical Volterra direction. These two properties a r e consistent only
which implies that U] satisfies the completeness equation
Using this equation we can rewrite the definition of the operator Hj in the form
which implies that Hj is self-adjoint since H0 and Wt commute. The case i = 2 is similarly considered. We now prove that the Hj a r e represented in the form
where the Vj a r e for multiplication by the functions
For this purpose, assuming again for the sake of definiteness that i = 1, we r e w r i t e Eq. (5.9) in the form
361
149 and take into account that the operator l', = I + A| is Voltcrra, so that
where 0(x) is the Heaviside function.
Equation (5.11) is rewritten in the form
where we set Vt = Ht — H0 and assume Vj to be an integral operator whose kernel can be a generalized func tion. The resulting equation is consistent with Vt being self-adjoint and U( being a Volterra only if
Moreover, we will see from this fact that A,(x, y) satisfies the partial differential equation
which, incidentally, we will not use. The operator H2 is analogously investigated. Our result is that the Hj a r e differential operators of the form
where the functions VJ(X) are given by Eqs. (5.10). Equation (5.11) also implies that the functions f,(x, k) and f2(x, k) constructed in t e r m s of the kernels of Aj(x, y) and A 2 (x, y) by means of Eqs. (1.8) and (1.9) are solutions of the differential equations
and have the asymptotics of Eqs. (1.2) and (1.3). Finally, the bounds of Eqs. (5.4), (5.5), (5.7), and (5.8) demonstrate that v t (x) and v2(x) satisfy bounds of the form
Moreover, if Fj(x) and F2(x) a r e n times differentiable, the potentials V](x) and v2(x) have n - 1 derivatives. Here v[ m J and v | m ' as x — » a n d as x « , respectively, behave as F [ m + 1 I and F J m + 1 ] . We conclude the study of the inverse problem by proving if fij(x) and fi2(x) are consistent, i.e., F t (x) and F 2 (x) a r e constructed in t e r m s of the given matrix S(k) satisfying the necessary properties given at the beginning of the section and m j 1 ' and mi 2 ' a r e related by Eq. (3.11), f,(x, k) and f,(x, k) satisfy the equations
It therefore follows that v t (x) and v2(x) coincide, and we obtain from the bounds of Eqs. (5.12) that
together with the corresponding refinements on differentiability in the case of the differentiability of F((x) and F 2 (x).
362
150 Wc have thereby found that the operator || = lln + V constructed by us belongs to the initial class of the Schrbedinger operators. Further, the equations (5.13) imply that H has the set of eigenfunctions u[ + ^(x, k) with asymptotic formulated in the table in Sec. 2, this asymptotic containing as the coefficient S]j(k) the matrix elements of the initial scattering matrix S(k). The construction of the operator H and thereby the proof of the sufficiency of the properties of the scattering matrix stated at the beginning of this section concludes with this fact. To prove Eqs. (5.13) we use the uniqueness theorem obtained above for the pair of functions u(x, k) and f(x, k), proving that the functions Uj(x, k) and fi(x, k) constructed using the Gel'fand-Levitan equations satisfy the equations
Equations (5.13) a r e thereby derived by using Eqs. (4.3) and (4.4). Suppose u 2 (x, k) and f2(x, k) a r e obtained using the equation
and the analyticity properties formulated above. We multiply Eq. (5.14) by s , , ( - k ) .
We obtain
The second t e r m in the left side is again replaced on the basis of Eq. (5.14),
In view of the unitarity condition of Eq. (2.3), the latter equation is rewritten in the form
We introduce the functions
The function u(x, k) is analytic everywhere in the upper half-plane except for the points k = i x j , where together with s,,(k) it has simple poles. The function f(x, k) has no singularities at k = ix?, since the singularities of u 2 (x, k) and s n ( k ) compensate each other. If s u ( 0 ) = 0, f(x, k) thereby lacks a singulari ty at k = 0. In fact if s,,(0) = 0, s12(0) = - 1 , and using Eq. (5.14) we find that u 2 (x, 0) = 0, so that the ratio u 2 / s n lacks a singularity at k = 0. Evidently, for real k,
We easily verify that the residues u(x, k) are related to f(x, k) by the equation
We need only use the condition of Eq. (3.11). Finally, Eq. (5.15) has the form in t e r m s of f(x, k) and u(x, k)
Based on the uniqueness of this pair of functions as formulated above, we conclude that
which implies, by Eq. (5.16), Eq. (5.13).
363
We conclude with this fact the general study of the inverse problem for the one-dimensional Schroedinger operator. 6.
Particular
Cases
of t h e
Solution
of t h e
Inverse
Problem
Here we will consider two examples where the inverse problem has an explicit solution: 1. Absence of reflection, i.e., the coefficients s 12 (k) and s 2 i(k) identically vanish and the entire nontrivial contribution to the G e l ' f a n d - L e v i t a n equation is provided by the discrete spectrum of H. 2. Rational reflection coefficient. Both examples a r e combined through the common property of the kernel Q(x + y) in the Gel'fandLevitan equation; it becomes degenerate and the solution reduces to q u a d r a t u r e s . However, we will con sider these examples separately. We will assume in the second example, in order to simplify the equations, that the discrete spectrum of H is absent. This will be sufficient for the reader himself to analyze, by combining methods for the first and second examples, which alternations to the equations arise in the general case. These examples make it possible to solve the inverse problem for a dense set of scattering data. Thus, let us consider the Gel'fand-Levitan equation (4.1) and assume that s12(k) - 0, so that the kernel ftj(x + y) has the form
The solution Aj(x, y) in this case is naturally found in the form
an algebraic system of equations naturally written in vector notation
arising for the function g/(x). Here g(x) is the desired column vector with components gj(x), I = 1 g0(x) is a column of the functions mj 1 '* - *'*, I = 1 , . . . , N and W,(x) is a matrix with elements
N,
The solvability of the resulting system is guaranteed by the general r e s u l t s of the preceding section and, solving it, it is possible to find gj(x) and, together with them, the k e r n e l A,(x, y). In particular, it can be easily verified that an expression is obtained for A](x, x) which in our notation can be written as follows:
We thereby find an expression for the desired potential,
We can s i m i l a r l y consider Eq. (4.2). We obtain for the potential v(x) the equation
where W2(x) is a matrix with elements
364
152 We note that the transmission coefficient in our case has the form
so that
and we recall that the constants m} 1 and m j 2 ' a r e related by the equation
It follows from the general considerations of Sec. 5 that Eqs. (6.1) and (6.2) for v(x) coincide under these conditions. Direct verification of this identity constitutes a nontrivial combinatorial problem. We now pass to the second example. Suppose s12(k) is a rational function of the variable k,
where P m (k) and Q n (k) are polynomials of degree m and n, m < n, having the identity element as coefficient for the highest degree k, and r is a constant. The r e a l n e s s condition
will hold if
where r 0 is a real number and the zeroes of the polynomials P m (k) and Q n (k) are located symmetrically about the imaginary axis. The constant r 0 must be sufficiently small in order that
For this purpose it is necessary that all the zeroes of Q n (k) have nonvanishing imaginary part. F o r s i m plicity we will assume that all these z e r o e s are simple. The case of multiple zeroes can be considered by the corresponding passage to the limit. The procedure described in Sec. 3 for reconstructing the transmission coefficient s n ( k ) in terms of S]2(k) can be explicitly performed. We recall that we have assumed that the discrete spectrum of H is absent, so that s n ( k ) has no poles in the upper half-plane. The explicit equation for Sn(k) has the form
where a and a}-> are the zeroes of Q n (k) in the upper and lower half-plane, respectively, and these a r e roots of the equation
in the upper half-plane. There are precisely n such r o o t s , since the equation is invariant relative to the substitution k k. Using Eq. (2.10), we can see that the second reflection coefficient s 2I (k) is also a r a tional function and is represented in the form
365
153 where the polynomials P m ((k) and Qn((k) and tne constant r< possess properties similar to that for P m , Q n , and r . We will also assume that all the zeroes of Qn, are simple. The Fourier transforms Ft(x) and Tz(x) of the reflection coefficients s,2(k) and sa,(k) are calculated using the Jordan lemma. In particular, Fiix)-j}p,e'a}+)x,
*>0;
»_ (-1
where the sums are taken over the zeroes of Qn(k) and Qn-(k) in the upper half-plane. The coefficients p, and p] coincide to within a factor i with the residues Sj2(k) and su(k) at the poles located at these zeroes. We will see that the kernels Ft(x + y) and F2(x + y) of the Gel'fand-Levitan equations (4.1) and (4.2) are degenerate when x > 0 and x < 0, respectively. We can use a method in these regions for solving them already mentioned in the investigation of the first example. As a result we find the expression for the desired potential *(jc)=-2^1ndet(/+Z1(jc)), *>0;
{$3)
^W=-2^1ndeH/+Z s W), x2'(x, k) of fj(x, k) and f,(x, k) for fixed x hnve an analytic continuation in the upper half-plane of the p a r a m e t e r k. For large k
For r e a l k the p a i r s of solutions f^x, k), f,(x, k) and f,(x, k), f2(x, k) form two fundamental s y s t e m s of solutions of Eq. (2.1). The transition coefficients a(k) and b(k) are introduced by the equations
These coefficients satisfy the identity
and can be expressed in t e r m s of fj(x, k) and f2(x, k) by means of the equation
Here {f, g}, an analog of the Wronskian, is defined as a bilinear form of the matrix
We will see that o(k) has an analytic continuation on the upper half-plane and is bounded there as k — °°,
The function a(k) does not have z e r o e s in the upper half-plane since such zeroes would correspond to com plex eigenvalues of the formally self-adjoint system (2.1). Knowledge of the fundamental system of solutions of the system of equations allows us to introduce solutions u(x, k) satisfying the radiation condition and to develop scattering theory similar to what was done in Sec. 2 Chap. 1 for the example of the Schrdedinger equation with decreasing potential. We will not carry this out here, limiting ourselves to stating that the matrix S(k) is determined by the transition coefficients a(k) and b(k) by means of equations that coincide letter for letter with those presented here. The sole difference is that the parameter k now runs through the entire real axis. In particular, the entire matrix S(k) can be reconstructed in terms of one of the reflection coefficients
We have for the solutions f^x, k) and f2(x, k) the integral representations
Here the column vector f0(x, k) has the form
and the kernels A,(x, y) and A 2 (x, y) a r e real matrix functions absolutely integrable with respect to y for fixed x. The kernels A,(x, y) and A 2 (x, y) can be used to construct the Volterra transformation o p e r a t o r s and, in p a r t i c u l a r , to express in t e r m s of them the completeness condition. We will not carry this out h e r e and p r e s e n t only equations expressing Q(x) in t e r m s of these kernels
372
160
and a formulation of one of the Gel'fand-Levitan equations
Here the matrix function F^x) is introduced by the equation
It is possible to prove, based on a study of this equation and its analog for the kernel A 2 (x, y), that the properties of S(k), viz., its unitarity and symmetry, the analyticity of the t r a n s m i s s i o n coefficient, and the absolute integrability of the Fourier transformations F,(x) and F,(x) of the reflection coefficients s 12 (k) and s21(k)
a r e necessary and sufficient properties of the scattering matrix for a canonical system with matrix Q(x) satisfying the condition of Eq. (2.2). We will now show how the canonical system is a generalization of the Schroedinger equation. If the matrix Q(x) has the form
the system of Eqs. (2.1) is equivalent to the Schroedinger equation with operator H of the form
which is evidently positively defined. That the equation
which also in fact reduces the range of variation of the p a r a m e t e r k to the s e m i - a x i s , is r e a l constitutes a necessary and sufficient condition on the scattering data corresponding to the matrix Q(x) of the form of Eq. (2.3). We note in conclusion that the case of the system (2.1), when the matrix Q(x) has the form
where m is a positive constant (system of Dirac equations with m a s s ) , was recently considered by I. S. Frolov [30]. 3. T r a c e
Formula
In this section we will derive identities relating the momenta of the function ln| a(k)|, where a(k) is one of the transition coefficients, with the integral of polynomials formed by derivatives of the potentials in the one-dimensional Schroedinger operator or in the operator of the canonical system. These equations first appeared apparently in [5], which in turn developed a paper by I. M. Gel'fand and V. M. Levitan [9]. In these works it was shown why such identities can naturally be called t r a c e formulas. The derivation which we present will not use general o p e r a t o r - t h e o r e t i c concepts and is taken from [10], We first consider the case of the Schroedinger operator. We will assume that the potential v(x) is a Schwartz-type function. Then the reflection coefficients s,,(k) and s 21 (k) d e c r e a s e as |*|-- 0. The equation 373
161
which constitutes one variant of Eq. (1.2.9) implies that In a(k) has a decomposition in negative degrees of k,
where
Even d e g r e e s vanish as a consequence of the evenness of the integrand ln|a(k)|. Bearing in mind the inter pretation of a(S\) as a regularized c h a r a c t e r i s t i c determinant of H, we can say that the C 2 J + 1 are propor2;+'
J
tional to regularized t r a c e s of the half-integer d e g r e e s H 2 of H, while such t r a c e s vanish for inte gral degrees. We now calculate the coefficients c n directly in t e r m s of v(x). This calculation can be interpreted as the definition of the matrix t r a c e of the o p e r a t o r s H 2 . Identities which are obtained by the setting equal of the thus obtained expressions for c n are called the trace formulas. This interpretation will not be evident in the elementary calculations which follow. We consider the function
It can be proved that this function is analytic in k when I m k > 0 and sufficiently large |k| for any fixed x. F o r l a r g e |x| it has the asymptotics
so that
where
The Schroedinger equation implies thatCT(X,k) satisfies the Riccati equation
which can be used to determine the asymptotic decomposition
We find for the coefficients c n (x) the recursion relations
374
162 The first few coefficients have the form
We will see that a 2 (x) and CT4(X) a r e total derivatives. This property is also preserved for all CTR(X) for even n. Returning to In a(k) we will verify that the coefficients c n in the decomposition of Eq. (3.1) are written in the form
so that, for example
We have thus found the set of identites
called the trace formulas. Their interpretation in terms of the traces of the half-integer degrees of H is particularly evident due to the presence in the right side of the sum of half-integer degrees of its discrete eigenvalues. The differential operator of the canonical system is considered analogously. We will use as the function o(x, k) following [11], the expression
It can be proved that
The canonical system (2.1) for CT(X, k) implies the equation
which can be used for the asymptotic decomposition of the type of Eq. (3.2). Here the first coefficients R. The trajectories of the Hamiltonian system are given by the differential equations
where $ i s a point of the manifold, | is a tangential vector to the trajectory at the point £, and J is the map of the 1-form in the vector fields defined by the form Q. (cf., for example, [2]). We now note that Eq. (4.1) can be written in the form
where the functional h[v] has the form
Comparing Eqs. (4.3) and (4.4) we see that the latter equation in fact is of Hamiltonian form, where h[v] plays the role of the Hamiltonian, while the antisymmetric operator J = 3/9x generates a simplectic struc ture in the set of functions v(x). The corresponding 2-form, which we write as a bilinear form of the vari ations 6{v and 62v of v(x) is determined by an operator inverse to J and having the form
376
164
The similar objects for Eq. (4.2) have the form
and
correspondingly. We see that the simplectic structure in the space of complex functions w(x) is induced by the natural complex structure of the real space of functions p(x) and q(x), where w = p + iq. The scattering problem is used to describe substitution of variables in Eqs. (4.1) and (4.2), under which they become explicitly solvable. We describe the corresponding scheme first for the example of Korteweg-de Vries equation. We consider the function v(x) as a potential in the Schrbedinger operator H studied in the preceding chapter. Suppose (S[2(k), KI, m/1*) a r e the corresponding scattering data which in turn uniquely define v(x). It turns out that the functional h[v] and the form Q cannot be expressed explicitly in t e r m s of the scattering data and their variations. The corresponding formulas have the form
and
where
the conversion factors a(k) and b(k) being constructed in t e r m s of the initial scattering data. We will see that the new variables are explicitly canonical, where P(k), p;, [ = 1 N play the role of canonical momenta, while Q(k), q/, Z = 1 , . . . , N play the role of canonical coordinates. Moreover, the Hamiltonian h[v] turns out to be a function only of the momenta, so that the Hamiltonians of the equa tion in new variables is trivially solved. Turning to the scattering data, the corresponding solution can be written in the form
These facts also make it possible to solve the Korteweg-de Vries equation using an auxiliary scat tering problem. Suppose v(x, 0) = v(x) is a Schwartz-type function defining the Cauchy data for this equati~" The sequence of maps
defines the corresponding solution. Here the first left a r r o w denotes the solution of the direct scattering problem for the Schroedinger equation with potential v(x) = v(x, 0), the next a r r o w is defined in Eq. (4.9), and finally, the last a r r o w constitutes the solution of the inverse problem.
377
165 It is precisely this proposition which is presented in the first work of Kruskal et al. [42]. The Hamiltonian interpretation presented here was obtained in [10]. Analogous formulas
and
where
can be found for the nonlinear equation of the second example. Here o(k) and b(k) are the conversion fac t o r s for the canonical system (2.1) constructed in t e r m s of the functions p(x) = Rew(x) and q(x) = Im w(x). We note that no factor k was present in the definition of the momentum P(k) and that the range of variation of the variable k is now the entire axis. These formulas were found in the thesis of L. A. Takhtadzhyan [20]. They imply that the general solution of the boundary-value problem for the equation is provided for by the sequence of maps
where
The s i m i l a r equation
(bearing in mind the opposite sign in front of the nonlinearity) was previously considered by V. E. Zakharov and A. B. Shabat [11]. The corresponding scattering problem is non-self-adjoint and has yet to be mathe matically investigated. Let us now dwell briefly on the derivation of Eq. (4.6), (4.7), (4.10), and (4.11). We consider only the Korteweg-de Vries equation, since the second example is considered analogously. Equation (4.6) has already been derived by us in the preceding section and constitutes the identity for the third t r a c e . In fact the coefficient c 5 is proportional to the functional h[v]. The right side of Eq. (4.6) a r i s e s if we bear in mind the definition of Eq. (4.8). We note that the preceding r e s u l t s imply that all the functional C2J+1 of the function v(x) p r e s e r v e their values for the Korteweg-de Vries equation. In fact the trace identities d e m o n s t r a t e that all these functionals depend only on momentum-type variables which do not vary with t i m e . This observation provides a simple and exhaustive approach to the description and completeness problem of the motion integrals for the Korteweg-de Vries equation, which has been dealt with in a broad l i t e r a t u r e . References can be found for example in [43]. To derive Eq. (4.7) for the form £2 we note that it is possible to obtain using the Gel'fand-Levitan equation an expression for the variation of the potential v(x) in t e r m s of the variations of the scattering data,
378
166 Here s ! 2 (k), f,(x, k), K J , and mi 1 ' are objects corresponding to the potential v(x) and f,(x, k) is the d e r i v a tive of fj(x, k) with respect to k. Calculation of the form Q is subsequently found by substituting this ex pression into Eq. (4.5) and calculating the integral with respect to x and y. For this purpose the following equation turns out to be useful
which follows simply from the Schroedinger equation. Details on the calculations can be found in [10], to which we refer the r e a d e r . By carrying out the calculations presented t h e r e , they can be easily carried over to the second example considered by us and Eq. (4.11) can be obtained. Equation (4.10) constitutes an identity for the t r a c e s No. 3 for the canonical system. With this we conclude the description of an unexpected application of the formalism of scattering theory for solving one-dimensional nonlinear equations. The inherent r e a s o n s why this scheme works as well as its range of application has yet to be clarified. One "experimental'' approach towards its use is to consider a one-dimensional differential operator for which the direct and inverse scattering problems are investigated and to describe the trace identities. Next one is to find a simplectic structure on the set of coefficients of this operator which can be explicitly expressed in t e r m s of the scattering data. Then, the equations generated by this structure and by some functional of the trace formulas as a Hamiltonian can turn out to be exactly solvable. Finally, this scheme has not been studied to any great extent, but it must be used for lack of a better scheme. Equation (4.2) was found precisely by this method. Searches for new scattering problems that can be studied are of particular interest. Generalizations of the Schroedinger equation or of the canonical system to the case of vector functions are obvious candi dates. The monograph [1] demonstrated that most results known for s c a l a r equations can be c a r r i e d over without any difficulties in the case of vector equations. Equations of higher o r d e r s however a r e m o r e p r o m ising. For example, V. E. Zakharov has recently proved that the t h i r d - o r d e r differential operator
generates the nonlinear equation
which plays the role of a continuum analog of the nonlinear Fermi—Pasta—Ulam problem. The inverse problem of scattering theory for the operator of Eq. (4.12) has yet to be solved. CHAPTER THREE-DIMENSIONAL
3
SCHROEDINGER
OPERATOR
In this chapter we will consider the Schroedinger operator in a space of functions depending on the three variables H=-/± +
v(x)=ff0+V.
Here *€R3, A is the Laplace operator, and v(x) is a function which we will assume to be r e a l , bounded, and sufficiently rapidly decreasing at infinity. Spectral analysis of this operator is significantly more complicated than the one-dimensional Schroedinger operator considered in Chap. 1. This particularly relates to the detailed investigation of such properties as continuity and the asymptotic behavior of the scattering amplitudes, which must be carried out in order to discuss sufficient or necessary conditions on this function corresponding to a potential v(x) of a given class. We will, therefore, in this chapter present only a formal scheme for solving the inverse problem, not making explicit each time under which conditions on v(x) do the discussions hold. A traditional condition on v(x) under which most of the bounds presented below hold asserts that
379
167
Under this condition the operator H defined in % = L7(R3) on the dense domain D = w'S(R3) is self-adjoint. We consider a three-dimensional case only for the sake of definiteness. All equations generalize without difficulty to the case of a r b i t r a r y n s 2. In appropriate places we will note analogies or divergences from the one-dimensional case treated in Chap. 1. The m a t e r i a l set forth first appeared in [29]. 1. S c a t t e r i n g
Theory
The construction of scattering theory for the pair of o p e r a t o r s H and H0 in a stationary variant is based on the existence of a set of solutions u'*'(x, k) of the Schroedinger equation
with the following asymptotic behavior at infinity, i.e., as |x| — °°,
(radiation condition). Here
The solutions uWfx, k) a r e s i m i l a r to the sets of solutions u j ^ x , k) and u ^ ( x , k) of Sec. 2 Chap. 1. The role of the index i = 1, 2 is played by the direction a = k/|k| of the vector k, which runs through the unit s p h e r e S2. An existence proof and an investigation of the solutions u ^ ' f x , k) was carried out by A. Ya. Povzner [18, 19]. The discussions presented in Chap. 1 and based on the existence of a fundamental system of solutions of the Schroedinger equation are not applicable to our case. Therefore, it is necessary to study directly the integral equations of scattering theory
Here G(±> (x, \k\) is G r e e n ' s function for the Helmholtz equation
which can be uniquely defined by the radiation condition. The explicit expression
well known in the three-dimensional case is obtained from the general equation
after calculation of the integral. In the last equation the well-known generalized function (x ± iO) * o c c u r s . The investigation of A. Ya. Povzner is based on Fredholm theory for Eqs. (1.3). An important role is played by the Kato theorem [35], which implies that homogeneous equations corresponding to Eq. (1.3) for real k do not have nontrivial bounded solutions. A. Ya. Povzner proved that the solutions u (±) (x, k) form a complete orthonormalized system of eigenfunctions of the continuous spectrum of H, which fills the e n t i r e positive s e m i - a x i s . This spectrum has uniform infinite multiplicity, so that the eigenfunctions a r e numbered in addition to the eigenvalue k2 by the point a 6S 2 . Besides a continuous spectrum, H can have a finite number of nonpositive eigenvalues of finite multiplicity. To simplify the equations we will assume that the entire discrete spectrum of H consists of a single negative eigenvalue, which we denote by - x 2 . The corresponding normalized eigenfunction, which can be assumed to be r e a l , is denoted by u(x).
380
168 The completeness and orthogonality conditions on the function u
