Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem

Acta Applicandae Mathematicae 48: 113±184, 1997. 113 c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

SERGIO ALBEVERIO1,2 and KONSTANTIN A. MAKAROV1,3 1FakultatÈ furÈ Mathematik, Ruhr UniversitatÈ Bochum, D-44780 Bochum, Germany 2BiBoS; SFB 237 Bochum-Essen-Dusseldorf;È CERFIM, Locarno, Italy 3Department of Computational and Mathematical Physics, St. Petersburg University, 198094 St. Petersburg, Russia

(Received: 29 January 1996)

Abstract. A model of a quantum mechanical system related to the three-body problem is studied. The model is de®ned in terms of a symmetric pseudo-differential operator (PDO) with unbounded symbol. The entire family of self-adjoint extensions of this operator is studied using harmonic analysis. A regularization procedure for this PDO is introduced and the spectral properties of the operators obtained in this way are investigated. The limit behavior of the regularized operators when the regularization parameter is removed is analyzed and a nontrivial attractor is exhibited. Mathematics Subject Classi®cations (1991). 47A20, 47B35, 81Q10. Key words: extension theory of symmetric operators, convolutions, pseudo-differential operators, harmonic analysis, E®mov effect, three-body quantum problem, regularization, attractors, Hamil- tonians which are not lower semibounded, spectral analysis, Lagrangian planes.

0. Introduction

In 1933, Wigner [1] remarked that nuclear forces must be of very short range and very strong. Two years later, Bethe and Peierls [2] in order to model such forces introduced point potentials that vanish except at the origin and so correspond to zero range forces. The introduction of zero-range potentials gave a powerful impulse to study quantum mechanical systems with singular interaction, many of which being also solvable [3]. In the monograph [4] one can ®nd an historical survey and a bibliography containing more than 500 references on this subject. This modelling was very successful in the area of one- and two-body systems with singular interactions located at centers in discrete points or even on certain continua of points. The extension of this type of models to the case of N-body systems [5] has met great dif®culties. The ®rst observation in this direction is due to L. H. Thomas [6] (in work of 1935), where he showed that in the three-body problem with short-range forces, the ground state of the energy operator can approach as the range tends to zero, even when the ground-state energies of all two-body−∞ subsystems remain constant. First work (1956) making use of zero-range potentials in the three-body systems is due to Skorniakov and

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Ter-Martirosian [7]. However the numerous attempts of numerical computation of the ground state of H3 or He3 in their model led to nonstable results and later it was noted by Danilov [8] that the integral equations derived in [7] to determine the wave-functions has nonunique solutions and thus the three-body system with point-like interactions needs for its full description some addition- al `experimental' parameter, which can not be computed using only two-body experimental data. Therefore at the begining of the 60's it was already clear that any correctly de®ned three-body Hamiltonian with point-like interactions involving only two-body interactions had to be unsemibounded from below (a phenomenon ®rst mentioned in [6]) and that there should be a one-parameter family of such Hamiltonians having the same two-body characteristics (as ®rst remarked in [8]). In 1961, Faddeev and Minlos [9, 10] published two marvellous short notes where the energy operator of three identical three-dimensional quantum particles (bosons) interacting in a ªpointed wayº, was studied. This operator was de®ned as a certain self-adjoint extension of the symmetric operator

H = ∆x ∆x ∆x (0.1) − 1 − 2 − 3 3 on the domain of functions of three variables x1,x2,x3 R that vanish whenev- ∈ er any two arguments coincide xi = xj,i=j, i, j = 1, 2, 3. Minlos and Faddeev have shown that the natural way to de®ne6 the operator corresponding to δ-like pair interactions proposed by Skorniakov and Ter-Martirosian [7] does not lead to a self-adjoint operator but only to a symmetric one (with de®ciency indices (1,1)) and therefore this operator has a one-parameter family of self-adjoint extensions. From the physical point of view, in order to ®x some self-adjoint extension of this operator, one needs to specify the behavior of the three-body wave functions in the neighborhood of the three-body collision point (where the particles are close to each other). Minlos and Faddeev found that all nontrivial self-adjoint extensions describing the energy operator have a discrete spectrum unbounded from below and therefore the corresponding quantum system with δ-like pair interactions collapse, i.e. we have a phenomenon of `fall to the center'. (In recent work [11], Minlos and Melnikov extended these results to the general case of three different particles of different masses). Thus in [9, 11], one can ®nd the mathematical `explanation' for the Thomas effect (unsemiboundedness of the energy operator from below) and also the interpretation of Danilov's `experimental' parameter as the one describing the one-parameter family of self-adjoint extensions of the initial symmetric operator (`pre-Hamiltonian'). The unbounded- ness from below of the Hamiltonian described above for a long-time diminished interest of the physicists in multiparticle models with point interactions. In fact when Faddeev in 1963 published his fundamental work [12] on the three-body problem with regular two-body forces, the interests of the physicists switched over to the numerical solution of the corresponding Faddeev equations

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.2 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 115 and the hope to ®nd simpli®cations in the three-body problem with δ-interactions was put aside. In 1977 Albeverio, Hoegh-Krohn and Streit pointed out [13] that if one allows in a three-body pre-Hamiltonian to have in addition to two-particle point interactions an effective three-particle interaction, then the Hamiltonian can be shown to be bounded from below. Such a Hamiltonian is then concretely given by a positive Dirichlet form. Apparently this observation was not further devel- oped, see however [5]. A stabilization of three-particle pre-Hamiltonians with point interactions was also obtained in another direction, namely by introducing internal degrees of freedom (see, e.g., [14]). Historically, the ®rst attempt to con- struct a semibounded energy operator for three-particle system with pair `δ-like interactions' (more precisely, δ-interactions for particles with internal degrees of freedom) was made by Shondin [15] and Thomas [16] (the original idea of such a regularization arose, however, earlier in work of Schrader [17] on a Lee-like model). In work [18] by Pavlov and in its modi®cation [19] by Makarov, one can ®nd the study of the general case of three-body Hamiltonians with generalized δ-like pair interactions which are semibounded from below. Concerning the original problem of a three-body Hamiltonian with two-body point interactions a new development was initiated by an apparently unrelated observation by the physicist E®mov who in [20] pointed out that three-body systems with short-range interactions may have an in®nite discrete spectrum, if the two-body subsystems have so-called zero-energy resonances. This fact, nowadays known as E®mov's effect, provoked a lot of investigations (see, e.g., [21±26] and references threin) and belongs to the most interesting phenomena in the theory of the multiparticle SchrodingerÈ operators. In his note, E®mov used some considerations based on Skorniakov±Ter-Martirosian ideas on zero-range interactions in the three-body problem. Some time later, Faddeev [27] draw the attention of Yafaev to the fact that Skorniakov±Ter-Martirosian equations and those one needs to prove in a rigorous way the E®mov's effect are of the same nature and in 1974 Yafaev published the mathematical theory of E®mov's effect [23]. For further mathematical work on the E®mov's effect, see [25, 26], the absence of the E®mov's effect in two dimensions for N-particle systems was shown recently in [28] by Dell'Antonio, Figari and Teta. The discussion on the E®mov's effect has been extended to the case of the discrete SchrodingerÈ equation [29] and its role in the nonrelativistic limit of quantum ®eld models has been analyzed (see, e.g., [30±32]). The next step was done in a letter by Albeverio, Hoegh-Krohn and Wu [24], where the close connection between E®mov's effect and the problem of three- body systems with point interactions was established. In this letter, the result was announced that if two-body potentials are spherically symmetric and of short range and if 2 or 3 of the two-body subsystems have a zero energy resonance, then the corresponding three-body system has in®nitely many spherically symmetric

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.3 116 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV states with energies En 0 such that limn En+1/En exists and only depends on the mass ratio and not→ on the potentials.→∞ The argumentation of this work was based on the study of scaled three-body Hamiltonians of the form

λα(ε) xα Hε = ∆+ 2 vα , (0.2) − α ε ε X where α = 1, 2, 3, numerates the pairs of particles, xα are relative Jacobi coordinates, λα(ε) are coupling constants and ε 0 is a small scaling parameter. In contrast to the two-body case [33, 34]→ where one can approximate Hamil- tonians with δ-interactions by means of local scaled short-range potentials, in the three-body case the Hamiltonian Hε does not converge at all as ε 0. And 1 → the ω-limit set (attractor) of the family of resolvents (Hε z)− as ε 0was conjectured to be the set of resolvents corresponding to the− one-parameter→ family of self-adjoint extensions of the Hamiltonian (0.1). Moreover, it was argued that 1 asymptotically the limit behavior of the resolvents (Hε z)− as ε 0isperi- odic in log ε. In addition, using scaling arguments Albeverio,− Hoegh-Krohn→ and Wu showed that the same arguments imply that asymptotically the behavior of the E®mov's series of eigenvalues tending to the threshold of the three-body system and the in®nite series of eigenvalues tending to in the Faddeev±Minlos model are connected by a simple scaling factor. −∞ In the present paper, we develop a mathematical theory suitable for the description of the complex phenomena described above. Our work can be considered as a ®rst step for the proof of all results announced in [24]. We plan to give application of the present work to the quantum three-body problem in subsequent work. We consider in the space L2(R) a pseudodifferential operator with unbounded symbol of the form = w(x)l( i )w(x), (0.3) A − ∇ where

α x w(x)=e2| |,α>0 (0.4) and l is a real-valued bounded function admitting an analytical continuation to the strip Π = z : Im z < 3 α (0.5) (3/2)α { | | 2 } andsuchthatlis bounded in this strip. The operator appears to be a correctly de®ned symmetric operator on the initial domain A( )= (W2),whereW is the multiplication operator by the function w. InD theA case whereD the function l vanishes at some points in the strip Πα/2 this operator has equal nonzero de®ciency indices and therefore admits a family of self-adjoint extensions. The

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®rst part of the paper (Sections 1±4) is devoted to the study of the corresponding self-adjoint extensions. The operator , more precisely, its block corresponding A to the subspace L2(R+) in the decomposition of L2(R) into the direct sum of its subspaces L2(R)=L2(R ) L2(R+), naturally arises in the theory of self-adjoint extensions for the three-body− ⊕ Hamiltonians (0.1) with δ-interactions [9, 11, 35±38]. The reduction of the study of the three-body quantum-mechanical problem to the study of such pseudo-differential operators actually constitutes a brilliant piece of mathematical physics, as can be read in the survey of Flamand [39] (see also [44], Ch. III). In the second part of the work Sections 5±10 we consider the special case where the function l has a limit at in®nity and we concentrate our attention to the study of having the following form A = W (1 )W, (0.6) A −L where is a self-adjoint integral operator with translation invariant kernel L: L f(x)= L(x y)f(y)dy. (0.7) L R − Z We suppose that the kernel L decreases exponentially at in®nity in such a manner that

β e |·|L( ) L (R) (0.8) · ∈ ∞ for some β>0. For some technical reasons (see Lemma 10.3), we also suppose that L(x) isaHolderÈ continuous function in x on the whole real axis. Along with the operator we also consider the one-parameter family of (unbounded) self-adjoint operatorsA

R = W (1 R)W, R > 0, (0.9) B −L 2 de®ned on the same domain ( R)= (W ),where R is an integral operator with the kernel D B D L x y L (x, y)=χ Lreg(x y)χ . (0.9a) R R R R − reg Here χ(x) means the characteristic function of the interval [ 1, 1] and LR is a periodic extension with period 2R of the function L from the− interval [ R, R] to the whole real axis. This family can be considered as some `regularization'− of a more complicated object . The main goal of the paperA is to study the attractor of the dynamical system R R in the sense of strong resolvent convergence like it has been discussed in→B [24]. It turns out that this attractor consists of a special family of self-adjoint extensions of the symmetric operator and we study precisely the corresponding dynamics in the neighborhood of theA attractor. Let us note that the introduction

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.5 118 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV of the cut-off parameter R corresponds to the problem of approximation of the three-body Hamiltonian with δ-interactions by means of scaled short-range local interactions. For our model we prove in a rigorous way the conjecture of the work [24] concerning the limit behavior in the strong resolvent sense of the three-body Hamiltonians Hε in (2) as ε 0. → Let us formulate the main results of the work.

THEOREM I. Let l be a real-valued function admitting an analytic continuation to the strip Π(3/2)α being bounded in this strip and satisfying the following conditions (i) The equation l(z)=0 (0.10)

has a ®nite number of solutions in the strip Π(3/2)α. 1 (ii) The inverse function l− is a bounded function in the neighborhood of in®nity in this strip. (iii) There are no solutions of (0.10) on the axis R i(α/2). ± Then the operator initially de®ned on the domain ( )= (W2)is a closed symmetric operatorA with equal de®ciency indices (n,D n)A,whereDnis a total number of zeros of (0.10) in the strip Πα/2 counting multiplicities.

It is useful to formulate some facts concerning the extensions theory of by passing to its Fourier transform. Let be a unitary equivalent operator toA : 1 A A = − ,where denotes the Fourier transformation. A FAF F b

THEOREMb II. The domain of de®nition of the adjoint operator ∗ can be represented in the form A b α α ( )= ( )+ p s+i +q s i ,p,q , (0.11) ∗ 2 2 D A D A − ∈Q where bis the spaceb of the rational functions q vanishing at in®nity and having poles onlyQ in the strip Π such that the function l(s)q(s),q is analytic in α/2 ∈Q the strip Πα/2.

The next assertions deal with the special case of the function l described in Theorem I of the form l(s)=1 L(s), (0.12) − where L the Fourierb transform of the kernel of the integral equation (0.7). Let us suppose that the real roots a1,a2,...,am,0

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.6 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 119 us denote by reg the subset of rational functions of the set which are regular Q Q on the real axis and let us denote by sing the linear subspace generated by the simple fractions Q 1 ,k=1,...,m. (0.13) s ak − + For a given rational function q reg let us denote by q (resp. q−) the only ∈Q + rational function from the regular subspace reg such that the function q q (resp. q q ) is an analytic function in the lowerQ (resp. upper) half-plane.− − − Let us consider a subspace ω of the de®ciency subspace ( ∗)/ ( ), parameterized by points ω of theDm-dimensional torus Tm D A D A b b ω =(eiα1 ,...,eiαm ) Tm. (0.14) ∈ We de®ne this subspace by α α = q+ s + i + q s i ,q + ω 2 − 2 reg D − ∈Q α α + q s i + q s + i ,q,q , (0.15) 1 2 2 2 1 2 sing − ∈Q provided that

iαk Res s=a q1(s)=e Res s=a q2(s),k=1,...,m. | k | k Two further main results of our work are:

THEOREM III. The restriction ω of the adjoint operator on the domain A ( ω)= ( )+ ω b (0.16) D A D A D is a self-adjointb operator.b

THEOREM IV. Let a1,...,am be simple real roots of the equation 1 = l(s) and let R ω(R) be the trajectory on the torus Tm → 2ia1R 2iamR ω(R)= (e− ,...,e− ). (0.17) − Under the condition β>4α,whereβis such that (0.8) holds, we have the following asymptotics in the strong resolvent sense

s lim Rz( R) Rz( ω(R)) = 0, Im z = 0. (0.18) − R { B − A } 6 →∞ Rz(A) denotes the resolvent of the operator A at the point z,i.e.Rz(A)= (A z) 1. − −

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Let us now give some more details on the content of the single sections in this work. In Section 1 the de®ciency indexes of the symmetric operator are computed and it is shown the is semibounded if and only if the function l Adoes not change A sign on the axis, i.e. l > 0orl60. In Section 2 we study the domain and the action rule of the adjoint operator to . AIn Section 3 we describe the family of all self-adjoint extensions of the operator . In particular, we use the theory of self-adjoint extensions of symmetric operatorsA having lacunas in their spectrum. In Section 4 we give a qualitative spectral analysis of the family of all self- adjoint extensions, we compute the position of the essential spectrum of these extensions and give some results concerning the behavior of the discrete spectrum. In Sections 5±8 using Fourier series, we study the asymptotics of the solutions (in some functional spaces) of the integral equation with the kernel LR given by (0.9a) R fR(x) LR(x, y)fR(y) dy = h(x) (0.19) − R Z− on the ®nite interval [ R, R] as R outside some neighborhood of the set of those values of the− parameter R→∞for which the corresponding homogeneous integral equation (0.19) has a nontrivial solution. The right-hand side h of (0.19) is supposed to be ®xed and to have compact support. In Section 9, the results obtained are applied to the study of the asymptotic behavior as R of the resolvents of the operators R in the sense of strong convergence. First→∞ we obtain these results at the pointB z = 0 of the values of the spectral parameter z (under the restriction that R outside some neighborhood of the set where the solution of the integral equation→∞ (0.17) is not unique) and then we extend these results to arbitrary complex values of z.We show that this asymptotics can be described in terms of the resolvents of the special subfamily of self-adjoint extensions ω(R) (from Theorem IV). In the last Section 10, using some traditionalA tricks of Krein's theory of singular perturbation of symmetric operators we prove the central result (Theorem IV) without any restriction on R passing to the limit R . →∞ 1. Description of the Domain of the Adjoint of the Basic Operator A

In this section we study unbounded operators in the Hilbert space L2(R) L2 given by the formal expression ≡ = w(x)l( i )w(x), (1.1) A − ∇ where l is a real bounded function and (α/2) x w(x)=e | |,α>0. (1.2)

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If the condition

l( i ) (W ) (W) (1.3) − ∇ D ⊂D holds, where (W )= f L2,Wf L2 is the domain of the multiplication operator W byD the function{ ∈ w,then∈ is} correctly de®ned by the expression (1.1) on the domain A

( )= (W2) (1.4) D A D as a symmetric operator. We show that if the function l admits an analytic bounded continuation to the strip Πα/2,where

Πα= z C, Im z <α , (1.5) { ∈ | | } then the condition (1.3) holds automatically. The operator commutes with the involution, being the composition of complex conjugationA and the re¯ection (f(x) f( x)), and therefore admits a family of self-adjoint extensions. As we shall7→ show− below (Theorem 1.2), if, in addition, the function l admits a bounded analytic continuation to the strip Π(3/2)α and has there a ®nite number of zeros then the symmetric operator has de®ciency indices (n, n),wherenis the total number of roots of the func-A tion l in the strip Πα/2 counting multiplicity. Moreover, the domain of the adjoint operator is complemented by the elements of the form A∗

( ∗)= ( )+Uα/2 + U (α/2) . (1.6) D A D A Q − Q

Here Uα is the operatione of complexe shift de®ned by

(eUαg)(s)=g(s+iα) (1.7) for anye meromorphic function g and is the space of rational functions (vanish- Q ing at in®nity) having singularities only in the strip Πα/2 such that the functions lq, q are analytic in the strip Πα/2. It∈Q is convenient to study the operator using the Fourier transform : A F L2(R) L2(R) → 1 ikx f = f(k)= e− f(x) dx. (1.8) F √ 2πZR b Let W be the operator which is unitary equivalent (by the Fourier transform) to the operator W c 1 W = W − . (1.9) F F c

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By some variant of the Payley±Wiener theorem (see, e.g., [40] Th. IX.13) the domain (W ) of W consists of functions f L2(R) admitting an analytic D ∈ α continuation to the strip Πα/2 such that for all η R, η 6 2 c c ∈ | | f( + iη) L2(R) (1.10) · ∈ and

sup f( + iη) L2 < . (1.10a) η α/2 || · || ∞ | |6

We call functions f with the property (1.10a) PW functions in the strip Πα/2. For later use we introduce the operators Uα of complex shift, for any α R. ∈ αx Let Uα be the self-adjoint multiplication operator by the function uα(x)=e b in the space L2(R) and de®ned on its natural domain. We call the operators Uα, acting in L2(R),de®nedby b 1 Uα= Uα − (1.11) F F operatorsb of complex shift.

Remark 1.1. The functions from the domain (Uα),α>0, admit an analytic continuation to the strip 0 < Im z<αwhich isD square integrable on every line parallel to the real axis in this strip. Then b

(Uαg)(s)=(Uαg)(s)=g(s+iα),g (Uα) (1.12) ∈D in theb sense of boundarye values of analytic functions.b Let us stress here the difference between the complex shift operator Uα which is de®ned as a self- adjoint operator in L2(R) and the operation of complex shift Uα, whose action is not limited to L2(R), see (1.7). Moreover, s inb (1.12) is real, whereas in (1.7) it is any value for which the meromorphic function g is de®ned.e

Let us denote by H the Hardy spaces, i.e. H+(H ) are the spaces of Fourier ± − transforms of L2-functions supported by the positive (negative) semiaxes. The operators W,U (α/2) are reduced by the Hardy spaces (in the terminology of, ± e.g., [41]). Moreover, the parts of W in H coincide with the parts of U (α/2) in H andc inb this sense the following representation± holds ± ± c b W = Uα/2P+ U (α/2)P , (1.13) ⊕ − − wherecP areb orthogonalb projectors into the Hardy spaces H . ± ± CONDITION (L1). Let L be the operator of multiplication by the real function l admitting an analytical continuation to the strip Πα/2 andsuchthatlis bounded in the strip Πα/2. We denote the analytic continuation of l by the same symbol.

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LEMMA 1.1. Under condition (L1) the operator = WLW is a symmetric operator on the domain A c c ( )= (W2). (1.14) D A D c Proof. Since the function l is an analytical bounded function, by the Payley± Wiener theorem and (1.10) we see that L (W) (W),i.e. is well de®ned. Let us verify the symmetry of .Letf,gD ⊂D( ), then (withA( , ) the scalar A ∈DA · · product in L2(R)) c c

( f,g)=(WLWf,g)=(LWf,Wg) A =(Wf,LWg)=(f,WLWg)=(f, g), (1.15) c c c c A since W = W con (Wc) , (Wc2) c (W),Lis a bounded self-adjoint ∗ D D ⊂D operator and LWg (W), as we have already remarked. 2 c c ∈D c c c Every symmetricc operatorc admits a closure. But under some more stringent assumptions on the function l we shall prove that is already closed on ( ). A D A CONDITION (L2). Let us assume that the boundary values of the function l on the boundary of the strip Πα/2 are bounded away from zero, in the sense that the multiplication operators by the functions l 1(s i α ) are bounded operators − ± 2 in L2(R).

LEMMA 1.2. Under conditions (L1) and (L2) is closed on ( ). A D A Proof. Since the Hardy spaces H are invariant for the operators W and ± U (α/2),wehave ± c b ( )= + ( ) ( ) (1.16) DA P D A ⊕P−D A and

P ( ) ( ) (1.17) ±D A ⊂D A and therefore by (1.13) the decomposition of L2(R)=H+ H leads to the following matrix representation for ⊕ − A A+ B+ = , (1.18) A B A ! − − where

A = U (α/2)P LU (α/2)P , ± ± ± ± ± (1.19) (A )=P ( ) D ± b ±D A b

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B = U (α/2)P LU (α/2)P , ∓ ± ± ∓ ∓ (1.20) (B )=P ( ). D ± b ∓D A b The operators B on (B ) are, however, bounded, since, say, for f (B+) we have ± D ± ∈D

B+f = Uα/2P+LU (α/2)P f − −

= Ubα/2P+U b(α/2)Lα/2P f = P+Lα/2P f, (1.21) − − − where Lα/2 isb the multiplicatione operator by the function α (L f)(s)=l s+i f(s), (1.22) α/2 2 which is a bounded operator due to condition (L1). Here we have used the property

P+Uα/2 = Uα/2P+. (1.23)

Analogouslyb b

B f = P L (α/2)P+f, f ( ). (1.24) − − − ∈D B− Therefore, the operators B can be extended as bounded operators acting in ± L2(R)

B = P L (α/2)P . (1.25) ± ± ± ∓ As a result of this extension we have

B+∗ = B (1.26) − by a symmetry principle (the function l is real on R, thus α α l s + i = l s i ). 2 2 − Therefore the difference diag(A+,A )is a bounded operator and, hence, A− − if diag(A+,A ) is closed, then is also closed on ( ). By (1.16) and (1.20) it is suf®cient− to prove that A Aare closed on (A D) separately.A ± D ± Let us prove that, say, A+ = A¯+ on (A+). Let fn (A+),fn n f, D ∈D → →∞ and A+fn n g in H+. By de®nition of a closed operator, we have to → →∞ check that f (A+)and A+f = g.SinceA+=P+UαL (α/2)P+ on (A+) ∈D − D and P UαL (α/2)P+ is a bounded operator, the convergence in L2(R) of the sequence− − b b UαL (α/2)P+fn = Lα/2UαP+fn − b b

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.12 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 125 is a consequence of the convergence in H+ of the sequence A+fn. Under condition (L2), the inverse of Lα/2 is a bounded operator and, hence, the sequence UαP+fn converges in H+. In turn, this means that P+f P+ (Uα)= (A+), ∈ D D since Uα is closed. Using now the boundedness of Lα/2 and repeating the rea- soningb in the opposite order one can easily see that A+f = g. b ¯ Theb proof of the property A = A follows the same scheme. 2 − − LEMMA 1.3. The domain of the adjoint operator is the orthogonal sum of A∗ the domains of A+∗ and A∗ considered as operators in H respectively: − ± ( ∗)= (A+∗) (A∗ ). D A D ⊕D − Proof. This result is an immediate consequence of the fact that the difference diag(A+,A ) is a bounded operator (we use here the matrix representation A− −

(1.18)). 2

The domains (A∗ ) are described by the following lemma. D ±

LEMMA 1.4. The domain of the adjoint operator A+∗ to the operator A+ in the Hardy space H+ is given by

(A+∗ )= f H+ :P+L (α/2)f (A+) . (1.27) D { ∈ − ∈D } Analogously

(A∗ )= f H :P Lα/2f (A ) (1.28) D − { ∈ − − ∈D − } in the space H . − Proof. The condition g (A )means that for all f (A+)the functional ∈D +∗ ∈D f (A+f,g) is bounded. Taking into account the series of equalities (where we→ use also (1.12)):

(A+f,g)=(Uα/2P+LUα/2P+f,g)=(P+Lα/2UαP+f,g)

=(UbαP+f,Lb (α/2)P+g)=(P+UαP+bf,L (α/2)P+g) − −

=(UbαP+f,P+L (α/2)P+g), b (1.29) − the condition of boundednessb of this functional means that

P+L (α/2)P+g (Uα H+)= (A+), (1.30) − ∈D | D since the operator Uα H isb self-adjoint in H+. | +

The proof of (1.28) is analogous. 2 b CONSEQUENCE 1.1. The condition g ( )holds if and only if ∈D A∗ P L P g (W2 ) (1.31) + (α/2) + H − ∈D | + c

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P L P g (W2 ). α/2 H − − ∈D | − Unfortunately (1.31)c does not give the possibility to obtain an explicit representation for the elements of the domain ( ∗). To obtain such a representation we need some further restrictions on theD functionA l.

CONDITION (L3). The function l admits an analytic bounded continuation to the strip Π 3 . 2 α

LEMMA 1.5. Under condition (L3) every element g ( )can be represented ∈D A∗ (in the strip Πα) in the form

f+(s) f (s) g(s)= + − , (1.32) l(s+iα) l(s iα) 2 − 2 for some functions

f (W2). (1.33) ± ∈D Proof.Letcg ( ). By Consequence 1.1 we have ∈D A∗ P L g (W2 ) (1.34) + (α/2) + H − ∈D | + and c

P L g (W2 ), (1.35) α/2 H − − ∈D | − where c

g = P g. (1.36) ± ±

It follows from (1.34) that the function P+L (α/2)g+ admits an analytic continu- − ation from the lower half-plane to the strip 0 6 Im z 6 α andisaPWfunctionin this strip. Since P L (α/2)g+ belongs to the PW class on the upper half-plane, − − and, in particular on the strip 0 6 Im z 6 α, then, as a consequence, L (α/2)g+ − belongs to the PW class on the strip 0 6 Im z 6 α. On the other hand, g+ belongs to the PW class on the lower half-plane, and, in particular, to the PW class on the strip α Im z 0. Since the function l is a bounded analytic − 6 6 function in the strip Π(3/2)α, we immediately conclude that L (α/2)g+ belongs − to the PW class on the strip α 6 Im z 6 0 also. Collecting these results, we 2− infer that L (α/2)g+ (W ). − ∈D 2 By similar arguments we also prove that Lα/2g+ (W ). c ∈D c

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In other words we have proved that one can ®nd some elements f (W2) such that ± ∈D c L (α/2)g+ = f and Lα/2g = f+. (1.37) − − −

This ends the proof of (1.32) and of Lemma 1.5. 2

Under condition (L2) we have thus proved the inclusion

( ∗) , (1.38) D A ⊂D where

1 2 1 2 = Lα/− 2 (W )+L−(α/2) (W ). D D − D

Of course, under Conditionc (L2) thec set is a subset of L2(R). Any function D from admits a meromorphic continuation to the strip Πα and may have poles D α there of ®nite order at the zeros of the functions l(s i 2 ) in the strip Πα. For simplicity of further considerations we assume± that the following addi- tional condition holds.

CONDITION (L4).

(i) The function l has a ®nite number of zeros in the strip Π(3/2)α. 1 (ii) The function l− is a bounded function in a neighborhood of in®nity in the strip Π(3/2)α. Now we are in a position to prove the main result of this section.

THEOREM 1.1. Under conditions (L2), (L3) and (L4) we have the following description for the domain of the adjoint operator A∗

( ∗)= ( )+Uα/2 + U (α/2) , (1.39) D A D A Q − Q where is the space ofe rationale functions (vanishing at in®nity) having poles only inQ the strip Π , such that the function l(s)q(s),q , is analytic in the α/2 ∈Q strip Πα/2. We recall that U (α/2) denotes the operation (1.7) of complex shift. Proof. First, let± us prove the inclusion (of the right-hand side of (1.39) in ⊃ the left-hand side).e For this it is suf®cient to prove that U (α/2) are subsets of ± Q ( ∗). Let, say, g = U (α/2)q,forsomeq . Let us prove that g ( ∗). D A − ∈Q e ∈DA It is clear that g has poles only in the strip 0 < Im z<αand, hence, g H+. By Consequence 1.1 ite is suf®cient to verify that ∈

P L g (W2 ) (1.40) + (α/2) H − ∈D | + c

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(the second Condition (1.31) holds automatically, since P g = 0). We note that −

P+L (α/2)g = P+L (α/2)U (α/2)q = P+U (α/2)lq. (1.41) − − − − By assumption, the function lq eis analytic in thee strip Π and therefore lq α/2 ∈ (W),sincelis a bounded function and the rational function q vanishes at in®nityD (this provides the property of square integrability of lq on every line parallelc to the real axes in the strip Π ). It follows P U lq (W2 ), α/2 + (α/2) H − ∈D | + which by (1.41) proves (1.40). e c The inclusion Uα/2 ( ∗)can be proved in a similar way. Let us now proveQ⊂D the oppositeA inclusion . Lemma 1.5 yields that every element g ( e )admits the representation⊂ (1.32). Subtracting the irregular ∈DA∗ parts of g at the zeros of the function l(s i(α/2)) in the strip Π(3/2)α under condition (L4) we get the representation ±

g = g + g+ + g , (1.42) − 2 where g e (W )and g are rational functions (vanishing at in®nity) having ∈D ± poles only in the strip Πα, such that the functions l(s (iα/2))g (s) are analytic ± ± in Πα. e c ˜ Let us be convinced that g Uα/2 + U(α/2) . We prove it, say, for g+, the other case being completely± similar.∈ Q Q Let the rational function g+ have a pole ofe multiplicity k at the point s0 in the strip 0 < Im z<αof the upper half-plane. Let us show that the function l has a root at the point (s0 i(α/2)) of multiplicity not less than k.Sinceg+ − has by assumption a pole at s0, the function P+g+ also has a pole at the point s0 of multiplicity k. Being an element of ( ∗), g+ satis®es the ®rst condition (1.31). But this condition holds only if theD functionA l(s i(α/2)) compensates − the pole of g+ at the point s0. In other words, the point (s0 i(α/2)) is a root of l of multiplicity not less than k. − If s0 is a pole of g+ of multiplicity k in the strip α

For the casee of thee function g the proof is analogous. 2 − We shall now prove that the operator has equal de®ciency indices and, therefore, admits a family of self-adjoint extensions.A

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THEOREM 1.2. The operator has de®ciency indices (n, n),where A n=dim (1.44) Q is the total number of roots of l in the strip Πα/2 counting multiplicity. Proof. By von Neumann's theorem it is suf®cient to prove that is a real operator. Let J be the operation of the complex conjugation. We haveA to prove that J ( )= ( ) (1.45) D A D A and J = J. (1.46) A A Recall that ( )= (W2)is the space of functions f admitting an analytic D A D continuation to the strip Πα which is square integrable on every line parallel to the real axis in this strip.c Since the function f(¯z) is an analytic continuation of the function f¯ (W2), all the properties of f mentioned above remain true, which proves (1.45).∈D To prove (1.46) letc us note that the operator L, being an operator of multiplication by a real function, commutes with J. Thus we only need to check the property

JW = WJ, (1.47) which isc a simplec consequence of the fact that after the Fourier transform (1.8) 1 − J = JU, (1.48) F F where U is the re¯ection operator (Uf)(x)=f( x) (1.49) − (α/2) x and the multiplication operator by the function w(x)=e | | commutes with

the involution JU,sincew(x)is an even function. 2

To obtain more information on ,itisusefulto®nda-priori semiboundedness conditions on it. A Let us consider the quadratic form a of the operator de®ned on the domain A [a]= ( )= (W2), (1.50) D D A D a[f]=( f,f). c (1.51) A In addition, let us also consider the extension a of the form a to the domain [a]= (W), (1.52) D D e e c

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a[f]=(LWf,Wf)= Wf 2(s)l(s)ds. (1.53) | | Z It is cleare that ac6 a,i.e.c c [a] [a] (1.54) D ⊂De and e a[f]=a[f],f [a]. (1.55) ∈D The semiboundedness conditions for the forms a and a are closely connected e with the property of whether the function l has real zeros or not. More precisely, let us consider three cases e Case (a). The function l is bounded away from zero on the real axis and inf l(s) > 0. (1.56) s R ∈ Case (b). The function l is bounded away from zero on the real axis and sup l(s) < 0. (1.57) s R ∈ Case (c). The (continuous) function l has values of different sign on the real axis and, therefore, has real zeros. We will consider cases (a) and (c) in details. Case (b) can be treated in a similar way. LEMMA 1.6. In case (a) the quadratic form a is semibounded from below and can be estimated from below as follows 2 e a[f] > inf l(s) f (1.58) s R || || ∈ (with e the norm in L2(R)).Incase(c) the form a is neither semibounded from below|| · || nor from above. Proof. The proof of (1.58) is based on the chain ofe inequalities

2 2 2 a[f]= Wf (s)l(s)ds > inf l(s) Wf > inf l(s) f . (1.59) | | s R || || s R || || Z ∈ ∈ c c 1 The lattere inequality holds since the operator W − is a contraction 1 W − 1 (1.60) || || 6 c (with c the operator norm). Let us consider case (c). Let|| us || ®rst prove that the form a is not semibounded from above. In case (c), one can ®nd a point s0 R such that l(s0) > 0. For given ε>0, let us ∈ introduce the function e 1 fε(s)= α . (1.61) (s s0)+i( +ε) − 2

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We have fε L2(R), fε is an analytic bounded function in the strip Π and ∈ α/2 therefore fε (W)= [a].Moreover ∈D D c 1 (Wfε)(s)= e , (1.62) (s s0)+iε − and forcε 0 we have the following asymptotic representation → l(s) π a[fε]= 2 2 ε 0 l(s0) (1.63) (s s0) +ε ∼→ ε Z − (in thee sense that the quotient of the right-hand side by the left-hand side tends to 1) and

2 ds 2π fε = 2 α 2 ε 0 l(s0). (1.64) || || (s s0) +( +ε) ∼ → α Z − 2 From (1.63), (1.64) we conclude that for any A>0 there exists an ε>0such that

2 a[fε] A fε . (1.65) > || || Thee proof that a is not semibounded from below follows in the same way, the only difference is that we choose the point s0, where the value of l is negative: l(s0) < 0. e 2

Now we are in a position to extend the result of Lemma 1.6 to the case of the quadratic form a

CONSEQUENCE 1.2. In case (a) the quadratic form a is semi-bounded from below

2 a[f] > inf l(s) f . (1.66) s R || || ∈ In case (b) a is semibounded from above. In case (c) the form a is neither semibounded from below nor from above.

Proof. The ®rst assertion (case (a)) is a consequence of the inequality a 6 a. Case (b) is proven in a similar way. In case (c) the statement that a is not semibounded is based on the following approximation result: for any f e [a], ∈D one can ®nd a sequence fn [a]such that e ∈D e fn f (1.67) → in L2(R) with the property

a[fn] a[f]. (1.68) → e

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In fact, in the representation where the operator W is the multiplication operator (α/2) x by the function e | |, it is suf®cient to put fn = χnf,whereχn is the indicator of the interval [ n, n].Forsuchfn we have the strong convergences − Wfn Wf and LW fn LW f in L2(R),sinceLis a bounded operator. This proves→ (1.68). → It follows from (1.65) that for any A>0, one can ®nd an f [a]such that ∈D a[f] A>0. e (1.69) f 2 > ||e || If fn f is a sequence satisfying (1.67) and (1.68), then → a[fn] lim 2 > A. (1.70) n fn →∞ || || This proves the assertion of Consequence 1.2. The proof of the unsemibounded-

ness from below follows a similar path. 2

CONSEQUENCE 1.3. In case (a) (resp. case (b)) all self-adjoint extensions of the symmetric operator are semibounded from below (resp. above). In case (c) all self-adjointA extensions of are neither semibounded from below nor from above. A

LEMMA 1.7. In the cases (a) and (b) the quadratic form a is the closure of the quadratic form a. Proof. Let us ®rst check that the form a is closed. Lete us note that the following two-sided estimate holds e 2 2 inf l(s) Wf 6 a[f] 6 sup l(s) Wf . (1.71) s ·|| || s ·|| || c c Hence, if fn f and thee sequence fn is a Cauchy sequence with respect to the metric generated→ by a, then the left inequality (1.71) proves that the sequence Wfn converges in L2(R). Since W is a self-adjoint operator, and therefore it is a closed operator, wee conclude that f (W)= [a]and c c ∈D D

lim Wfn = Wf. c n e →∞

The right inequalityc c (1.71) proves the convergence fn f with respect to the metric generated by a, which proves the closability of a→. The properties (1.67), (1.68) show that a is the minimal closed extension of a, i.e. the closure a ecoincides with a (in the semibounded cases (a) or (b), of

course). e 2 e

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2. General Properties of the Adjoint Operator Throughout this section we assume that the function l satis®es conditions (L2), (L3) and (L4). Let us describe the action of the adjoint operator ∗ on ( ∗). Theorem 1.1 implies that every element g ( A)can beD A represented as ∈D A∗ g = g + g+ + g , (2.1) − where e g ( ) (2.2) ∈D A and e g+ = U (α/2)q+, (2.3) −

g = Ueα/2q (2.4) − − for some q . ±e∈Q THEOREM 2.1. Let g ( )be represented by (2.1)±(2.4),then ∈D A∗ ∗g= g+Wlq+ +Wlq . (2.5) A A − Proof.Since cis closed,c it is suf®cient to verify that eA ∗U (α/2)q+ = Wlq+ (2.6) A − and e c ∗Uα/2q+ = Wlq . (2.7) A − Let us check (2.6). The proof of (2.7) is analogous. e c By de®nition of the space , the element g+ = U (α/2)q+ belongs to the Q − Hardy space H+ and, hence, P+g+ = g+. Then for f ( )by (1.18) and (1.26) we have e ∈DA

( f,g+)=( f,P+g+)=(P+ f,g+) A A A =(A+P+f+B+P f,g+)=(f,A+∗ g+ + B g+). (2.8) − − The result of this computation together with (1.24) shows that

A+∗ g+ + B g+ = P+UαP+L (α/2)g+ + P L (α/2)g+ − − − −

= Uα/b2P+lq+ +U (α/2)P lq+ =Wlq+. (2.9) − − Here we used the inclusion lq+ (W), which holds by the de®nition of the b ∈D b c space . 2 Q c The next results describe the structure of the kernel of ∗ and the fact that the mapping f fis a surjection from ( ) onto theA whole Hilbert space. →A∗ D A∗

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LEMMA 2.1. The kernel of has the form A∗ Ker ∗ = Uα/2q U (α/2)q,q , (2.10) A { − − ∈Q} and therefore its dimensione e equals the total number of zeros of the function l in the strip Πα/2 counting multiplicities. Proof. The fact that the right-hand side of (2.10) is a subset of Ker ∗ is an immediate consequence of (2.6) and (2.7). Let us now prove that everyA element g Ker ∗ admits the representation (2.10). With the notations (2.1)±(2.4) the equality∈ A g = 0 means that A∗

g = W (lq+ + lq ). (2.11) A − − 1 Actinge on (2.11)c from the left by the bounded operator W − and taking into account that by de®nition = WLW, we infer that A c LWg = l(q+ + q ), c c (2.12) − − and, therefore,ce we have the representation

Wg = (q+ + q ). (2.13) − − It is clearc that W g (W),sinceby(2.2)g ( )= (W2).This equality e ∈D ∈DA D shows that the function Wg admits an analytic continuation into the strip Πα/2. c c c On the other hand,e by (2.13) the function Weg is an element of the space .By Q the de®nition of the spacece , this is only possible in the case where Wg = 0, i.e. Q ce ce q+ = q , (2.14) − −

which proves (2.10). 2

LEMMA 2.2. The image of the operator coincides with the space L2(R) A∗ Ran ∗ = L2(R). (2.15) A Proof. Let us prove the solvability of the equation

∗f = h, (2.16) A 1 f ( ∗),forallh L2(R).For such h we have W − h (W)and, ∈DA ∈ 1 ∈D therefore, the function 1/l(s)(W − h)(s) is meromorphic in the strip Πα/2. Under c c Condition 4, one can ®nd a rational function qh , such that c ∈Q 1 1 W − h qh (W). (2.17) l − ∈D c c

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Let 1 f = W 1 W 1h q (2.18) 1 − l − h − and c c

f2 = U (α/2)qh. (2.19) − Using (2.17)e and (2.18), we conclude that the element f1 belongs to the domain of the symmetric operator A 2 f1 (W )= ( ) (2.20) ∈D D A and c

f2 ( ∗). (2.21) ∈D A Setting

f = f1 + f2 (2.22) and applying the result of Theorem 2.1, we get

∗f = f1 + ∗f2 A A A 1 = WLWW 1 W 1h q +Wlq = h (2.23) − l − h h −

which ends thec proofcc of the lemma.c c 2

Remark 2.1. The solutions of (2.16) can be determined up to solutions of the corresponding homogeneous equation. In fact, in the proof of the lemma, we could choose instead of (2.19) the element

f2 = U+(α/2)qh. (2.24)

Later we shalle need some special solutions of (2.16) which look more `symmetric'. The rational function qh appearing in (2.17) can be represented as

Mk m ak qh(s)= m. (2.25) (s sk) Xk mX=1 − Here sk are the zeros of the function l and 0 6 Mk 6 deg sk, where deg sk are + their multiplicities. Let us introduce the functions qh and qh−:

Mk am 1 Mk am q+(s)= k + k (2.26) h (s s )m 2 (s s )m m= k m= k k:ImXsk>0 X1 − k:ImXsk=0 X1 −

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+ q−(s)=qh(s) q (s). (2.27) h − h

The poles of qh± correspond to poles of the function qh at the closed upper and, resp. lower, half-planes with the same Laurent's coef®cients at these (complex, non)(real) poles as the ones of qh. Then the element 1 f = W 1 W 1h q +(U q+ + U q ) (2.28) − l − h (α/2) h α/2 h− − − is also a solutionc ofc the inhomogeneouse equation (2.16),e since

+ (U (α/2)qh + Uα/2qh−) U (α/2)qh Ker ∗. (2.29) − − − ∈ A Let use also note thate if the functione l does not have real zeros, then the element f from (2.28) can also be represented as

1 1 1 f = W − L− W − h. (2.30)

c c m Remark 2.2. Laurent's coef®cients ak from (2.25) of the rational function qh 1 are determined by the values of the derivatives of the function W − h at the zeros sk of the function l. The following estimate c dm 1 iskp k (α/2) p (W − h) s=s = e− (ip) e− | |h(p) dp dsm | k Z

c b 1/2 2k α p 2 p e− | | dp h(p) dp 6 · | | Z Z = Cm(α) h , b (2.31) || || where h is the Fourier transform of h and Cm are some constants independent k on h, shows that these coef®cients am(h) are bounded functionals of h. b

3. Description of the Self-Adjoint Extensions of A We have already proved that the symmetric operator has equal de®ciency indices (n, n),wheren=dim (Theorem 1.2). TheA description of all self- adjoint extensions of is reducedQ according to the general theory to the search of maximal (n-dimensional)A subspaces in the complement ( )/ ( ) of D D A∗ D A ( ) in ( ∗),where ∗ is a symmetric operator. These subspaces are in one toD oneA correspondenceD A withA the Lagrangian planes of the symplectic form of the operator A∗ (f,g)=[f,g]=( ∗f,g) (f, ∗g),f,g ( ∗)/ ( ). (3.1) E A − A ∈D A D A

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We call an n-dimensional subspace of the 2n-dimensional space ( ∗)/ ( ) a Lagrangian subspace, if it is a null-subspaceD of the form . D A D A To describe such subspaces , it is useful to transformE to the standard n D E canonical form. Let pi,qi i=1 be a canonical basis in the de®ciency subspace ( )/ ( ): { } D A∗ D A

[pi,pk]=0, [qi,qk]=0,i,k=1,...,n, (3.2) [pi,qk]=δik.

Let ξ+(f) and ξ (f), ξ (f) Cn,betwon-dimensional vectors of the coor- − ± ∈ dinates of the element f ( ∗)/ ( )with respect to the canonical basis (3.2) ∈DA DA

n + f = ξi (f)pi + ξ−(f)qi. (3.3) i= X1

Then in terms of the variables ξ± the quadratic form can be written in the standard symplectic form E

+ + (f,g)=(ξ (f),ξ−(g)) n (ξ−(f),ξ (g)) n (3.4) E C − C and an n-dimensional plane is a Lagrangian plane if and only if the linear manifold D

+ n n (ξ (f),ξ−(f)) C C ,f { ∈ × ∈D} is the graph of some self-adjoint relation in Cn. Thus, the problem of the description of all self-adjoint extensions of is reduced to the search for some canonical basis (3.2) in the de®ciency subspaceA ( ∗)/ ( ). After that all self-adjoint extensions of can be parametrized by self-adjointD A D A relations in the space Cn. A First, let us ®nd the canonical basis in the simplest case where the function l has only simple zeros in the strip Πα/2. For every simple zero a of the function l in the strip Πα/2 let us de®ne two functions 1 f +(s)= , (3.5) a s (a+iα ) − 2 1 f −(s)= . (3.6) a s (a iα ) − − 2 + It is clear that the collection fa ,fa− a:l(a)=0 is a basis in the de®ciency subspace ( )/ ( ). { } D A∗ D A

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THEOREM 3.1. The collection pa,qa ,where { }a:l(a)=0 + fa +fa− pa= , (3.7) 2 πl0(a) p+ fa¯ fa¯− qa = − . (3.8) 2i πl0(¯a) is a canonicalp basis. (Here by √z we denote the principal branch of the square root: Im √z>0for Im z>0.) Proof. The assertion of the theorem is a consequence of the following relations: for all a, b,wehave + [fa ,fb−]=0. (3.9) If a = ¯b,then 6 + + [fa ,fb ]=0, [fa−,fb−]=0. (3.10) For every pair of complex conjugate zeros a and a¯,wehave + + [f ,f ]= 2πil0(a), (3.11) a a¯ − [fa−,fa¯−]=2πil0(a). (3.12) In fact by (3.9) we have 1 1 [p ,p ]= [f++f ,f+ +f ] a b 4π a a− b b− l0(a) l0(b) 1 p 1p = ([f +,f+]+[f ,f ]). (3.13) 4π a b a− b− l0(a) l0(b) ¯ If a = b, then [pa,ppb]=0p by (3.10). Otherwise the equality [pa,pb]=0isa consequence6 of (3.11) and (3.12). In a similar way, we have

[qa,qb]=0. (3.14) It is also clear that (3.10) implies

[pa,qb]=0 (3.15) for a = b. To calculate only the nontrivial terms [pa,qa], we apply (3.11) and (3.12):6

1 1 + + [pa,qa]= ([fa + fa−,fa¯ fa¯−]) 4π( i) l(a) l(¯a) − − 0 0 1 p +p+ = ([fa ,fa¯ ] [fa−,fa¯−]) = 1. (3.16) −4πil0(a) −

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Thus the only thing we need is to verify the relations (3.9)±(3.12). Let us prove (3.9). By the de®nition of the functions fa± (see (3.5) and (3.6)), we infer that

fa± H . (3.17) ∈ ± Applying the result of Theorem 2.1, we conclude that

+ 1 1 ( ∗fa ,fb−)= Wl , iα A ( a) ( b+ )! ·− ·− 2 c 1 1 = (Uα/2P+ + U (α/2)P )l , iα . (3.18) − − ( a) ( b+ )! ·− ·− 2 b b Since by (3.17) fa− H , one can continue (3.18) in the following way: ∈ − + 1 1 ( ∗fa ,fb−)= U (α/2)P l , iα A − − ( a) ( b+ )! ·− ·− 2 b 1 1 = P U (α/2)l , iα − − ( a) ( b+ )! ·− ·− 2 b 1 1 = U (α/2)l , iα − ( a) ( b+ )! ·− ·− 2 b α 1 1 = l s i α α ds R − 2 (s a i ) (s ¯b i ) Z − − 2 − − 2 l(z) = dz. (3.19) R i(α/2) (z a)(z ¯b) Z − − − Analogous calculations show that

+ l(z) (fa , ∗fb−)= dz, (3.20) A R i(α/2) (z a)(z ¯b) Z − − − which, together with (3.19), proves (3.9). To prove (3.10)±(3.12) we note that

+ + 1 1 ( ∗f ,f )= Wl , A a b ( a) ( b iα) ·− ·− − 2 ! c l(z) = dz (3.21) R+i(α/2) (z a)(z ¯b) Z − − and

+ + 1 1 (f , ∗f )= ,Wl a A b ( a iα) ( b) ·− − 2 ·− ! c

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l(z) = dz. (3.22) R i(α/2) (z a)(z ¯b) Z − − − Subtracting (3.22) from (3.21), we get a contour integral which can be explicitly computed by the residue theorem

+ + l(z) [f ,f ]= 2πiResz=a . (3.23) a b − (z a)(z ¯b) − − For the case a = ¯b, the function l(z)/((z a)(z ¯b)) is analytic in the strip 6 − − Πα/2 and therefore the right-hand side of (3.23) equals zero, which proves the ®rst equality (3.10). If a = ¯b, then (3.11) is an immediate consequence of (3.23).

The proof of the second equality (3.10) and (3.12) follows in a similar way. 2

In the case of arbitrary multiplicity of the zeros of the function l, the con- struction of the canonical basis needs some slight modi®cation. For every zero a of the function l with multiplicity m, let us de®ne the functions 1 f +,k(s)= , (3.24) a (s (a+iα))k − 2 1 f ,k(s)= , (3.25) a− (s (a iα))k − − 2 for k = 1,...,m. The proof of the following assertion is completely analogous to the one of (3.9)±(3.12).

LEMMA 3.1. For all a, b, 1 6 k 6 deg a, and 1 6 l 6 deg b, we have +,k ,l [fa ,fb− ]=0. (3.26) If a = ¯b, then 6 +,k +,l [fa ,fb ]=0 (3.27) and

,k ,l [fa− ,fb− ]=0. For every pair of complex conjugate zeros a and a¯ we have

+,k +,l l(z) [f ,f ]= 2πiResz=a , (3.28) a a¯ − (z a)k+l −

,k ,l l(z) [f ,f− ]=2πiRes . (3.29) a− a¯ z=a (z a)k+l −

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This lemma shows that in order to give to (3.1) the standard symplectic form (3.4) it is suf®cient to know the matrix S(a) at every zero a of multiplicity m of the function l, given by its matrix elements l(z) S (a)=Res ,k,l m, (3.30) kl z=a (z a)k+l 6 − since the only nontrivial terms are of the form

+,k +,l [f ,f ]= 2πiSkl (3.31) a a¯ − and ,k ,l [fa− ,fa¯− ]=2πiSkl. (3.32) This matrix S(a) is obviously symmetric, i.e.

S†(a)=S(a), (3.33) but, in general, it is not Hermitian (i.e. S∗(a) = S(a). In addition, by a symmetry principle for the function l (l is a real function6 on the real axis) these matrices corresponding to conjugate zeros are adjoint to each other

S∗(a)=S(¯a). (3.34) In particular, for real zeros a of the function l, these matrices are Hermitian, i.e. S∗(a)=S(a). ,k m It is useful to pass to a new basis ga± k=1 instead of the one given by (3.24), (3.25) for every zero a of multiplicity{ }m of the function l separately. Let U(a) be some nondegenerate m m-matrix and let × ,k ,l ga± = Ukl(a)fa± (3.35) Xl be the elements of a new basis on the linear span of (3.24), (3.25). By the de®nition (3.35), the elements of this basis are still satisfying conditions (3.26) and (3.27). Equations (3.31), (3.32), rewritten in terms of this new basic elements, become +,k +,l [g ,g ]= 2πiGkl(a), (3.36) a a¯ − ,k ,l [ga− ,ga¯− ]=2πiGkl(a), (3.37) where the matrix G(a)= G(a) k,l is of the type { } G(a)=U(a)S(a)U∗(¯a). (3.38) We note that the property (3.34) remains true for the matrix G(a)

G∗(a)=G(¯a). (3.39)

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The trick is now to select matrices U(a) and U(¯a) in such a manner that the matrix G(a) in (3.38) becomes diagonal. The selection procedure should be done simultaneously for a and a¯. That this is possible is a consequence of the following lemma, as we shall see below.

m LEMMA 3.2. For every m m-matrix S there is a basis xi i=1 of its eigenvectors and (in the general case× ) of its adjoint vectors and a{ biorthogonal} basis y m consisting of eigenvectors and (in the general case) of adjoint vectors { i}i=1 of the adjoint matrix S∗ such that

(Sxi, yj)=δijλj, (3.40) where λi,i=1,...,m, are eigenvalues of S counting multiplicity.

m Now let a be a zero of the function l of multiplicity m.Let xi(a) i=1 and y (a) m be basis of eigenvectors and adjoint vectors of the matrices{ S}(a) and { i }i=1 S∗(a)=S(¯a) from Lemma 3.2. Let us de®ne the matrices U(a) and U(¯a) as l U(a)kl = xk, (3.41)

l U(¯a)kl = yk, (3.42)

l l where xi(yi) is the lth component of the vector xi(a) (yi(a)). Then by (3.38), the matrix elements of the matrix G(a) are of the form

l m j m G(a)ij = xiSlm(a)yj = Sml(a)xi yj Xl,m Xl,m =(S(a)xi,yj)=δijλj(a), (3.43) which means that this matrix has the diagonal form

G(a)=diag(λ1(a),...,λm(a)), (3.44) where λ1(a),...,λm(a), are eigenvalues of the matrix S(a) counting multiplicity. We remark that by (3.39) the matrix G(¯a) is also a diagonal matrix ¯ ¯ G(¯a)=diag(λ1(a),...,λm(a)). (3.45) Let us also note that the matrix S(a) is nondegenerate. This follows from the following properties, which are a direct consequence of (3.30):

Skl(a)=0fork+l6m (3.46) and

(m) Skl(a)=l (a)=0fork+l=m+1, (3.47) 6

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(wehaveassumedthatais a zero of multiplicity m). By (3.46) and (3.47), we immediately conclude that det S(a)=( 1)ml(m)(a)=0, (3.48) − 6 which proves that S(a) is nondegenerate. Next, letting as before (see Theorem 3.1)

+,k ,k k ga + ga− pa = , (3.49) 2 πλk(a)

+p,k ,k k ga¯ ga¯− qa = − , (3.50) 2i πλk(¯a) it is easy to checkp that the basis (3.49), (3.50) is a canonical basis.

We end this section by describing two special self-adjoint extensions of . One of them, which we denote by , corresponds to the restriction of theA adjoint operator to the domain A− A∗ ( )= ( )+Ker ∗. (3.51) D A− D A A The operator is obviously a self-adjoint operator, since the subspace Ker ∗ is the LagrangianA− plane of the form (see Lemma 2.1). A The result of Lemma 2.2 means,E in particular, that there is at least one self- adjoint extension + of the operator which has a bounded inverse. Equation (2.28) suggests howA the domain of suchA an extension looks like. + For every rational function q,q , let us de®ne two functions q and q− by (2.26), (2.27) using the decomposition∈Q (2.25).

THEOREM 3.2. The restriction + of the adjoint operator to the domain A A∗ + ( +)= ( )+ U (α/2)q + Uα/2q−,q (3.52) D A D A { − ∈Q} is a self-adjoint operator. e e Proof. Let us check that the subspace

+ = U (α/2)q + Uα/2q−,q (3.53) D { − ∈Q} +,k is a Lagrangiane plane of thee form . Obviously dim = n.Letf and xa ,k E D ∈D and x−a be the coordinates of the element f with respect to the basis (3.24), (3.25). By de®nition of the subspace , the following relations are satis®ed: D +,k xa = 0, for Im a<0, ,k xa− = 0, for Im a>0, (3.54) +,k ,k xa = xa− , for Im a = 0.

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Then, the canonical coordinates ξ±(f) from (3.3) corresponding to the canonical basis (3.49), (3.50) satisfy the linear relations

+,k ,k ξ = iξ− , for Im a>0, a − a¯ +,k ,k ξa = iξa¯− , for Im a>0, (3.55) ,k ξa− = 0, for Im a = 0. In matrix form one can rewrite these relations as + ξ− = Cξ , (3.56) where C = C∗ is some Hermitian matrix (we do not specify its matrix elements). + Therefore the pair (ξ ,ξ−) belongs to the graph of some self-adjoint operator, which is a particular case of a self-adjoint relation. This means that + is a A self-adjoint operator. 2

THEOREM 3.3. The self-adjoint operator + given by (3.52) has a bounded inverse. Its resolvent at the point zero has theA form 0 0 1 1 1 1 0 U (α/2)qh + Uα/2qh − h = W − W − h q + − , (3.57) A+ l − h 2 e e where q0 is thec only functionc from having only real poles and such that h Q 1 1 0 W − h q L2(R). (3.58) l − h ∈ c Proof. Equation (2.28) proves that Ran + = L2(R). By the closed graph A theorem, + has a bounded inverse, and the action of the inverse operator is given accordingA to (2.28) as follows: 1 1h = W 1 W 1h q +(U q+ + U q ), (3.59) +− − l − h (α/2) h α/2 h− A − − where qh is suchc a functionc that e e e e e

1 1 eW − h qh (W). (3.60) l − ∈D If qh hasc a representatione c analogous to (2.25) (we use here and in what follows the same notations) e Mk am q (s)= k , (3.61) h (s s )m m= k Xk X1 − we pute

Mk am q0(s)= k (3.62) h (s s )m m= k k:ImXsk=0 X1 −

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0 and then qh q L2(R) and it is easy to see that − h ∈ U q0 + U q0 1e 0 + (α/2) h α/2 h W − (qh qh)=(U (α/2)qh + Uα/2qh−) − , (3.63) − − − 2 e e which proves (3.57). 2 c e e e e e CONSEQUENCE 3.1. The point zero is an eigenvalue of of multiplicity n. A− Proof.Since +has a bounded inverse, some neighborhood of zero does A O not contain the spectrum of +. The operators + and are self-adjoint extensions of the same symmetricA operator , whichA has ®niteA− de®ciency indices (n, n). As a consequence, this neighborhoodA does not contain more than O n eigenvalues of . On the other hand, Ker ∗ Ker , and therefore A− A ⊂ A− dim = n. 2 A− Let us note that in case (a) (resp. in case (b)), the function l has no real roots and hence the symmetric operator is semibounded from below. In this case, A the extension + coincides with the Friedrichs extension of . A A To prove this let us check that + de®ned by (3.52) corresponds (in this case) to the quadratic form a (1.52), (1.53).A By Theorem 3.3, it is clear that in case (a) the resolvent of + is of the form A 1 1 1 e 1 − = W − L− W − . (3.64) A+ If f ( +)and g [a], then the element f can be represented as ∈D A c ∈Dc 1 1 1 f = W − L− W − h (3.65) e for some h L2(R) and c∈ c 1 1 ( +f,g)=(h, g)=(WW− h, g)=(W− h, Wg) A 1 1 =(LL− W −ch,cWg)=(LWf,c Wg)=c a(f,g), (3.66) which means that + corresponds to the quadratic form a. By Lemma 1.7 a is A c c c c e the closure of a and therefore + is the Friedrichs extension of . A A e e 1 1 1 If the function l has real zeros, then the operator W − L− W − is not bounded and therefore it is not directly related to the resolvent of the operator +. Nevertheless, the following regularization theorem holds.c c A

THEOREM 3.4. The resolvent of the operator + at the point zero can be represented in the form A 1 1 1 1 1 1 1 W − Lε− W − + W − L−εW − +− = s lim − , (3.67) A − ε +0 2 → c c c c where Lε is the multiplication operator by the function l(s + iε)

(Lεf)(s)=l(s+iε)f(s).

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1 1 1 Proof. We shall prove that the family of operators W − Lε− W − is uniformly bounded with respect to ε for ε (0,ε0]and ε0 small enough and that the strong | |∈ convergence (3.67) takes place on a dense set in L2(Rc). c By condition (L4) (i) the function l(s + iε) does not vanish on the real axis (for ε small enough) and by condition (L4) (ii), we infer that the function 1 1 1 1 l− (s+iε) is bounded on R. Therefore, the operators W − Lε− W − are bounded if ε (0,ε0]. | |∈ c c1 1 1 Let us check the uniformly boundedness of the family W − Lε− W − .Let m(s)be a rational function having poles only at real zeros of the function l such that the function c c 1 n(s)= m(s) (3.68) l(s)− is analytic in the strip Πε0 .LetNε and Mε be multiplication operators by the functions n(s + iε) and m(s + iε), respectively. The family Nε is uniformly bounded and due to the equality 1 1 1 1 1 1 1 W − Lε− W − = W − NεW − + W − MεW − , (3.69) 1 1 it is suf®cientc toc provec the uniformlyc boundednessc c of the family W − MεW − . 1 Letus®xsomeelementf L2(R).The function W − f (W)admits an ∈ ∈Dc c analytic continuation into the strip Πα/2. Let us put c c q (f)(s)=[m(s+iε)(W 1f)(s)] , (3.70) ε − Πε0 where the brackets [g] denote the singular parts of the meromorphic function g Πε c in the strip Πε. Then the function qε(f) is a rational function having poles only at the points sk iε,wheresk are real zeros of l. The Laurent coef®cients of the − function qε(f) at its poles can be expressed by the Laurent coef®cients of the 1 function m(s) at sk and by derivatives of the function W − f at the points sk iε. As a consequence of Remark 2.2, these coef®cients are bounded functionals− of f. Let us prove that the family of operators Bε c 1 Bεf = MεW − f qε(f) (3.71) − is uniformly bounded.c In fact, let be some neighborhood of the set of real zeros of the function l. Outside theO set the function m(s + iε) is uniformly bounded. In addition, the following estimateO holds (uniformly in ε)

χR qε(f) 6 C f , (3.72) || \O || || || where χ∆ denotes the indicator of the set ∆. (We recall that the Laurent coef®- cients of qε(f) are bounded functionals of f.) In the neighborhood of the real 1 O zeros of l the function MεW − f qε(f) is continuous and bounded and admits the following estimate (see Remark− 2.2) uniformly with respect to ε c 1 1 MεW − f qε(f) ( ) 6 W − f k( ) 6 C f , (3.73) || − ||C O || ||C O || || c c

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.34 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 147 for k large enough. Here k( ) denotes the space of k-times differentiable functions on equipped byC theO natural norm. O 1 1 To end the proof of the uniform boundedness of the family W − MεW − ,we use the representation c c 1 1 1 1 1 W − MεW − f = W − (MεW − f qε(f)) + W − qε(f) − and thec estimatec c c c 1 W − qε(f) C f (3.74) || || 6 || || 1 which holdsc since the function W − qε(f) has poles at the points sk iε i(α/2), the Laurent coef®cients at these points being bounded functionals− of f±. Next let us check the strongc convergence (3.67) on the dense set of rational functions. Let h L2(R) be a rational function. We have the representation ∈ 1 1 1 1 1 1 1 1 W − L− W − h = W − (L− Mε)W − h + W − MεW − h. (3.75) ε ε − The functionc nc(s + iε) de®nedc by (3.68) isc uniformlyc boundedc with respect to ε and converges to the function n(s) uniformly on every compact set. This proves the existence of the limit

1 1 1 1 1 1 1 lim W − (Lε− Mε)W − h = W − W − h mW − h . ε 0 − l − → c c c1 c1 c If h is a rational function, then W − MεW − h is also a rational function and 1 1 the existence of limε +0 W − MεW − h can be checked by direct computation. → c c 1 1 1 Thus, we have proved that the limit limε +0 W − Lε− W − h exists. To compute c c → 1 this limit let us note that one can replace the function MεW − h in (3.75) by the 1 c c function [MεW − h]Πε , not changing the result. As above the brackets denote the 1 c irregular part of the function MεW − h in the strip Πε.Forε>0 the rational c 1 function [MεW − h]Πε belongs to the Hardy class H and converges to the 0 c − rational function qh from Theorem 3.3, the convergence being uniform on every compact set onc the real axis which does not contain real zeros of the function l. This implies that

1 1 1 1 1 1 0 0 lim W − Lε− W − h = W − W − h qh + Uα/2qh. (3.76) ε +0 l − → The computationc ofc the limitc expressionc for ε 0 followse in the same way. →− 1 Now, the only difference is that the function [MεW − h]Πε is an element of the Hardy class H+ and, hence, c 1 1 1 1 1 1 0 0 lim W − Lε− W − h = W − W − h qh + U (α/2)qh. (3.77) ε 0 l − − →−

Finally (3.67)c followsc from (3.76)c andc (3.77). e 2

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4. Qualitative Spectral Analysis of the Extension + A The position of the essential spectrum of the operator + is basically determined by the behavior of the function l(s) at in®nity. TheA simplest case, which is, however, the most important for applications, is the one when the function l(s) has a limit at in®nity.

CONDITION (L5). The following limit exists 1 lim l(s)= >0. (4.1) s a →±∞ The following assertion is based on a classical result (see, e.g., [40, Th. XI.20]) concerning the compactness of operators which are compositions of a multiplication operator by a function, resp. a multiplication operator by a function in the Fourier representation. Let 2 q< and 6 ∞ n f,g Lq(R ), (4.2) ∈ then

f(x)g( i ) Σq. (4.3) − ∇ ∈ We recall that Σq denotes the set of compact operators B such that B q = || || (tr B q)1/q < . | | ∞ LEMMA 4.1. If 1 a Lq(R [ R, R]),q2, (4.4) l(s) − ∈ \ − > for some R, then the resolvent of + is a compact perturbation of the operator 2 A aW − ,i.e.

1 2 c +− aW− Σ . (4.5) A − ∈ ∞ Proof. In thec case where the function l does not have real zeros the proof of (4.5) is very simple. In fact in this case we have

1 1 1 1 2 1 1 1 − = W − L− W − = aW − + W − (L− a)W − . (4.6) A+ − Here the propertyc (1/lc(s)) a cLq(R)cis a consequencec of (4.4), since l does − ∈ 1 α x not vanish on the real axis. In addition, e− 2 | | Lq(R) for every q>0.By 1 1 ∈ 1 1 1 (4.2) we infer that (L− a)W − Σq and therefore W − (L− a)W − Σq, 1 − ∈ − ∈ since W − is a bounded operator, which together with (4.6) proves the assertion. c c c c

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In the case where the function l has real zeros, the resolvent of + can be represented by Theorem 3.3 as A 1 1h = aW 2h + W 1 a W 1h q0 + Kh, (4.7) +− − − l − h A − − where we use thec notationsc of Theoremc 3.3 and K is the ®nite rank operator given by

0 0 U (α/2)qh + Uα/2qh Kh = − . (4.8) 2 e e 1 2 Let us prove that the operator −+ aW − K is compact. We ®rst have to isolate the singularities of the functionA − 1/l. Let− be some neighborhood of the real zeros of l and χ be the indicator ofc the setO . By repeating the arguments used in the proof ofO Theorem 3.4, we infer that inO the norm of the space C( ) of continuous functions on , we have the estimate O O 1 1 0 1 W h q W h k C h . (4.9) l − h 6 − C ( ) 6 L2(R) − C( ) || || O || || O c c Taking into account the inequality

1/2 χ f 6 (mes ) f || O || O || || with the help of (4.9) we obtain the estimate

1 1 1 0 1/2 W − χ a W − h qh 6 (mes ) C h L (R), (4.10) O l − − O || || 2 L2(R)

c c e for some constant C not depending on the set for mes small enough. Let us prove that the operator O O e 1 1 1 0 h W − (1 χ ) a W − h qh (4.11) O l → − − − is compact.c It is clear that the operatorc

1 0 h W − (1 χ )qh (4.12) → − O is of ®nitec rank. Then the compactness of the operator (4.11) is a consequence of (4.2), since by (4.4) the function (1 χ )((1/l(s)) a) is an element of the space − O − 1 2 Lq(R). Thus we have proven that for every ε>0, the difference +− aW − can be represented as the sum of a compact operator and one of normA less− than ε (one can choose the set in such a manner that the norm of the operator (4.10)c O becomes arbitrarily small). This proves assertion (4.5). 2

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Remark 4.1. If the function l does not have real zeros we can obtain a stronger result. Suppose that 1 a Lδ(R),δ>1, (4.13) l − ∈ 2 2 δ 2 δ/2 (we recall that L2(R) is the set of functions f such that (1+x ) f(x) L2(R) < ). Then the difference || || ∞ 1 2 − aW − Σ1 (4.14) A+ − ∈ is a trace classc operator. α x δ In fact, the function e− | | belongs to L2 (for any δ>0) and then (4.14) is an immediate consequence of Th. XI.21 [42]. Using standard arguments of perturbation theory, we are now able to formulate the central result of this section.

THEOREM 4.1. Under condition (4.4) the essential spectrum of + ®lls the interval [ 1 , ) A a ∞ 1 σ ( )= , . (4.15) ess + a A ∞ Outside this interval, the spectrum of + is discrete with possible accumulation 1 A point a− . In the semibounded case (i.e. if the function l does not have real zeros),the operator + would have a ®nite number of negative eigenvalues. If the function l A changes the sign on the axis, then + has a in®nite series of negative eigenvalues going to . A Proof.−∞ The assertion of the theorem is a consequence of the spectral theorem and the result of Lemma 4.1. In fact, the resolvent of the operator + is a compact 2 A perturbation of the operator aW − , which has an absolutely continuous spectrum in the interval [0,a] of multiplicity two. But under compact perturbations, the essential spectrum remains invariantc

1 2 σess( − )=σess(aW − ), A+ 1 which proves (4.15). Outsidec the interval [0,a], the spectrum of +− is discrete having possible accumulation points 0 and a. In the case whereA l does not have real zeros, + is semibounded and therefore the point zero cannot be A 1 an accumulation point of eigenvalues of +− . If l changes the sign, then + is unbounded from below and, hence, zeroA has to be an accumulation pointA of 1 eigenvalues for − . Applying the spectral theorem once more, we prove the A+ remaining assertions of the theorem. 2

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Remark 4.2. In the semibounded case (under Condition (4.13)) one can prove 1 2 that + has a part which is unitary equivalent to the operator a− W , since in this caseA we can apply the results of trace class perturbation theory. c Remark 4.3. The results of Theorem 4.1 remain true for all self-adjoint extensions of the symmetric operator . A

5. Integral Equation on the Axis

Let us consider on the space L2(R) the integral operator with the kernel depending on the difference of the arguments L

( f)(x)= L(x x0)f(x0)dx0,f L2(R). (5.1) L R − ∈ Z We assume that the kernel L is symmetric, i.e. L(x)=L¯( x), and that it exponentially decreases at in®nity, i.e. −

β e |·|L( ) L (R) (5.2) · ∈ ∞ for some β>0. Let l(s)=L(s)be the Fourier transform of the kernel L.Then the function l(s) is real and admits an analytic continuation into the strip Πβ. The operator is boundedb and after Fourier transform becomes the multiplication operatorL by the function l(s) and therefore its spectrumL coincides with the interval

σ( )= inf l(s), sup l(s) . (5.3) L "s R s R # ∈ ∈

On the space L2(R), let us consider the integral equation of the second kind (1 )f =h. (5.4) −L If 1 / σ( ), then the operator (1 ) has a bounded inverse and Equation (5.4) ∈ L −L 1 has a unique solution for every h L2(R). Otherwise, the operator (1 ) is ∈ −L − not bounded and Equation (5.4) has solutions for a dense set of h L2(R) only. Let us suppose that the equation ∈ l(s)=1 (5.5) has a ®nite number of real roots ak,k = 1,...,m. In this case, 1 σ( ) and the corresponding homogeneous Equation (5.4) has a family of solutions∈ L iakx m e k=1 in the space L (R). Let us note that in the case of degenerate roots, { } ∞ ia x the homogeneous equation has solutions of the type Pk(x) e k ,wherePk is a polynomial whose degree coincides with the multiplicity of the root ak minus one.

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Along with (5.4) in the space L2(R) we consider the `regularized' equation

(1 R)f =h, (5.6) −L where the regularization operator R was de®ned in (0.9a) and in notations of (0.9a) has the kernel L x y L (x, y)=χ Lreg(x y)χ . (5.7) R R R R − Obviously is compact, as any integral operator with a bounded kernel on a ®nite interval,L and self-adjoint. Hence, Equation (5.6) is of the Fredholm type and if we choose the cut-off parameter R in such a manner that

1 / σ( R), (5.8) ∈ L then Equation (5.6) has only one solution in L2(R). In the case where Equation (5.5) does not have real roots we shall see that for R large enough Condition (5.8) holds and

1 1 s lim (1 R)− =(1 )− . (5.9) − R −L −L →∞ In contrast, when Equation (5.5) has real solutions one can ®nd arbitrary large values of the parameter R such that the corresponding homogeneous Equation (5.6) has a nontrivial solution and therefore the operator (1 R) does not have a bounded inverse. In this case, the behavior of the solutions−L of (5.6) as R →∞ outside the set R :1/σ( R) becomes strongly irregular and any attempt to { ∈ L } describe such a behavior in the space L2(R) (for arbitrary but ®xed right-hand side h L2(R)) seems to be hopeless. Nevertheless, we will study this behavior ∈ for R outside some neighborhood of the set Z = R :1 σ( R) in →∞ { ∈ L } some weighted spaces L2,δ when the right-hand side h has compact support. Here we de®ne the spaces L2,δ as the spaces consisting of all function f such δ that exp( )f() L2(R). 2 |·| · ∈ We shall proceed as follows. For ®xed h L2(R), the solution of the integral ∈ Equations (5.6) is reduced to the inversion of the operator (1 R) in the space − L L2([ R, R]),where R is the integral operator in the space L2([ R, R]) given by the− same kernel asL in (5.7) e − e

R = PR R L2([ R,R]), (5.10) L L | − whereePR is the orthogonal projection operator from the space L2(R) onto the subspace L2([ R, R]). Thus (5.6) is reduced to the solution of the problem − (1 R)fR = h (5.11) − L in this space L2([ R, R]). Here we denote by the same letter the element h e − ∈ L2(R) and its restriction to the interval [ R, R] (being considered as an element −

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.40 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 153 of L2([ R, R]) ). Under the condition that Equation (5.11) has a solution fR − it is easy to compute fR using Fourier series. It appears that the leading term for R of the asymptotic of the Fourier series for the solution fR (under the condition→∞ that the right-hand side h has compact support) coincides with the Riemannian integral sum corresponding to a partition of the interval of integration by the points πn/R n Z for the integral { } ∈ 1 h(s) eips ds. (5.12) 2π R 1 l(s) Z b − Let us note that when the function l takes the value 1 the corresponding integrand in (5.12) has a nonintegrable singularity. Nevertheless, the asymptotic behavior for R for such Riemannian sums can be computed explicitly. After such →∞ a computation, it becomes possible to obtain the asymptotics of the solution fR of (5.11) and, resp. (5.6), in the weighted spaces L2,γ. Let us study the integral operator R acting in the space L2([ R, R]).The decomposition by the Fourier series correspondingL to the family− of orthogonal exponents on the interval [ R, R] e − 1 ei(πn/R)x (5.13) √2R n Z ∈ de®nes a unitary mapping U from the space L2([ R, R]) onto the space l2 of − sequences. In this representation, the operator R is diagonal L 1 U RU − = diag(...,ln,...), (5.14) L e where e R i(πn/R)x ln(R)= L(x)e− dx (5.15) R Z− are the Fourier coef®cients of the kernel L(x) with respect to the system (5.13). Therefore the spectrum of R coincides with the set of the Fourier coef®cients L ln(R) n Z of the function L, more precisely { } ∈ e σ( R)= n Z ln(R) . (5.16) L ∪ ∈ { } Under thee Conditions (5.2) on the function L(x) we have LEMMA 5.1. For R uniformly on n Z the following asymptotic representation holds →∞ ∈ πn l (R)=l + (e βR),R , (5.17) n R − O →∞ where l(s) is the Fourier transform of the kernel L(x)

isx l(s)= L(x)e− dx. ZR

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CONSEQUENCE 5.1. If (5.5) does not have real roots, then the operator (1 − R) has a bounded inverse for R large enough. L

e Consequence 5.1 shows in particular that the spectrum of R is rather `dense' in the interval (5.3). It is also clear that when the function 1L l(s) takes values of different sign on the axis, then there are arbitrary large valuese− of the `cut-off' parameter R such that the homogeneous integral equation (1 R)f = 0has − L nontrivial solutions in the space L2([ R, R]). Recall that we denote by Z the− set of values of R such thate the operator (1 R) has a nontrivial kernel − L

Ze = R :1 σ( R) . (5.18) { ∈ L }

Let us assume that Equatione (5.5) has a ®nite number of real solutions ak,k = 1,...,m,and l(0) = 1. It is clear that the set Z is a subset of some neighborhood 6 m of the union of a ®nite number of lattices = k=1 n Z πn/ak .Amore precise information is in the following Z ∪ ∪ ∈ { }

LEMMA 5.2. Let ZN (resp. N ) be the intersection of the set Z (resp. ) with the interval [N,N + 1] Z Z

ZN = Z [N,N + 1], ∩ (5.19) N = [N,N + 1]. Z Z∩ Then there is some constant C>0such that, for N large enough, the following inclusion is true

(β/p)N ZN R:dist(R, N ) CN e− , (5.20) ⊂{ Z 6 } where p is the maximum multiplicity of the roots ak,k =1,...,m, and β is the constant entering condition (5.2). Proof.Letus®xsomen Z.WhenRis a root of the equation ∈

ln(R)=1, (5.21) then using (5.17) we conclude that

πn βR x: 1 l(x) Ae− , (5.22) R ∈{ | − |6 } where A is a positive constant. Hence, there is a value of k,16k6msuch that

πn pk a B e βR, (5.23) R k 6 − −

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.42 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 155 where pk is the multiplicity of the root ak and B is some another constant, which is a simple combination of A and the value of the pkth derivative of the function l at the point ak. It follows from (5.23) that πn B1/pk R R e (β/pk)R. (5.24) a 6 a − k − k Taking into account that we are interested in the values R from the interval [N,N + 1] only and maximizing the estimate (5.24) with respect to k, k =

1,...,m, we get (5.20). 2

For the values of R outside the set Z the operator (1 R) has a bounded inverse and the solution of (5.11) can be represented by the− FourierL series (which converges in the space L2([ R, R])) e − 1 h (R) f (x)= n ei(πn/R)x, (5.25) R 2R 1 l (R) n Z n X∈ − where R i(πn/R)x hn(R)= h(x)e− dx (5.26) R Z− are the Fourier coef®cients of the function h. In the case where h has compact support we have the representation πn h (R)=h , (5.27) n R when R is largeb enough, where

isx h(s)= h(x)e− dx (5.28) ZR is justb the Fourier transform of the function h. Before going over to the study of the asymptotic behavior for R of the →∞ solution fR of (5.25) we need some simple results concerning Fourier series. Let the function h have compact support. On the interval [ R, R] let us consider the Fourier series for h corresponding to the exponents (5.13)− 1 h (R) ei(πn/R)x, (5.29) 2R n n Z X∈ where hn(R) are the Fourier coef®cients (5.26). LEMMA 5.3. Let h have a compact support, then for R uniformly on M, we have the asymptotic representation →∞ M 1 2 1 2 1 hn(R) = h (s) ds + M . (5.30) 2R | | 2π M | | O R n 6(XRM/π) Z− | | b

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Proof. Let us note that the left-hand side of (5.30) is just the integral sum for the integral from the right-hand side of (5.30). Since h has compact support, h0 L (R) and therefore the difference between the integral and the corresponding∈ ∞ b integral sum can be estimated from above by the value (πM/R) h0 ,which || ||∞ proves (5.30). 2 b The above result allows us to estimate the tails of the Fourier series (5.29).

LEMMA 5.4. There is a function ξ(M)

lim ξ(M)=0, M →∞ such that the following inequality holds

1 2 lim hn(R) 6 ξ(M). (5.31) R 2R | | →∞ n >(RM/π) | | X

Proof. Applying the result of Lemma 5.3 to the Parseval equality

R 1 2 2 hn(R) = h(s) dx, (5.32) 2R | | R | | n Z Z X∈ − we have 1 h (R) 2 2R n n >(RM/π) | | | | X R 2 1 2 = h(x) dx hn(R) R | | − 2R | | Z n (RM/π) − | |6X R 1 M 1 = h(x) 2 dx h(s) 2 ds + M . (5.33) R | | − 2π M | | O R Z− Z− Let ε>0. For M large enough, we haveb the estimate 1 ε h(s) 2 ds< . (5.34) 2π s >M | | 2 Z| | b At the same time for R large enough, we have ε h(x) 2 dx< (5.35) x >R | | 4 Z| |

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.44 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 157 and the remainder in (5.30) can also be estimated as less or equal ε/4. This proves the assertion of the lemma, since the following equality holds: 1 h(x) 2 dx = h(s) 2 ds. (5.36) | | 2π | | ZR ZR

(because of the unitarity of theb Fourier transform). 2

6. Integral Equation on a Finite Interval

Let us turn back to the study of the solution fR (5.25) of the integral equation (5.11) in the space L2([ R, R]). Let us decompose the Fourier series (5.25) into − two parts corresponding to the sets of summation index n : n 6 MR/π resp. n : n >MR/π : { | | } { | | } M tails fR(x)=fR (x)+fR (x), (6.1) where 1 h (R) f M (x)= n ei(πn/R)x (6.2) R 2R 1 l (R) n (MR/π) n | |6X − and 1 h (R) f tails(x)= n ei(πn/R)x. (6.3) R 2R 1 l (R) n>(MR/π) n | | X − Using the asymptotics (5.17) and representation (5.27), we see that the leading M term of fR for R has the same behavior as the integral sum corresponding to the partition of→∞ the interval [ M,M] by the points πn/R, n MR/π, − | | 6 1 h(πn) IM(x)= R ei(πn/R)x, (6.4) R 2R 1 l(πn) n (MR/π) R | |6X −b for the nonconvergent integral

1 M h(s) eisx ds. (6.5) 2π M 1 l(s) Z− −b The `tails' (6.3) can be estimated in an appropriate way using the result of Lemma 5.4. Hence, one can hope that the asymptotic behavior of the solution fR can be described in terms of the asymptotics of the integral sums (6.4) for the nonconvergent integrals (6.5). We also note that the integrand in (6.5) has m nonintegrable singularities like the poles 1/(s ak) , 1 m deg ak, where − 6 6 ak are real roots of the equation 1 = l(s) on the interval [ M,M]. Let us give precise formulations of these arguments. −

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LEMMA 6.1. Let ξ(M) be the function given by Lemma 5.4. We have the estimate

tails 1 lim fR L2([ R,R]) 6 sup 1 l(s) − ξ(M). (6.6) R || || − s >M | − | →∞ | | q Proof. The assertion is an immediate consequence of (5.31) and the fact that the function 1 l(s) is bounded away from zero for s >M,forMis large | − | | | enough. 2

The estimate on the difference between the series (6.2) and the integral sum IM depends on the distance d(R) from the set πn/R,n Z to the set R { ∈ } a1,...,am { } πn d(R)=dist a ,...,a , ,n Z , (6.7) 1 m R { } ∈ where a1,...,am, are the real roots of the equation l(s)=1.

LEMMA 6.2. Let a function h have compact support and let p be the maximum βR multiplicity of the roots a1,...,am. Then outside a (e− )-neighborhood of m O the set = k=1 n Z πn/ak for suf®ciently large R we have the estimate Z ∪ ∪ ∈ { } M M √ 1 βR fR IR L2([ R,R]) 6 C(h) MD− (R)e− , (6.8) || − || − where

p p βR D(R)=d (R)(d (R) (e− )) (6.9) −O and C(h) is some constant only depending on h. Proof. Using (5.17) and (5.27), let us estimate the difference between the M M Fourier coef®cients for the function fR (6.2) and the integral sum IR (6.4). Since the function h(s) is bounded, we have

πn hn(R) b h( R ) πn 1 ln(R) − 1 l( ) − −b R

πn 1 πn πn = (1 l (R)) 1 l − h l (R) l n R R n R − − −

1 βR b = πn (e − ). (6.10) (1 ln(R))(1 l( ) O | − − R | We need some bound for the denominator in (6.10). Let us ®x some neighborhood of the roots a1,...,am.Fornsuch that πn/R / the value (1 ln(R))(1 Ol(πn/R) is bounded away from zero by some constant∈O which does| − not depend− |

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.46 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 159 on R for R large enough. For the values of n such that πn/R ,wehavethe estimate ∈O πn 1 l Cdp(R), (6.11) R > − where d(R) is de®ned by (6.7) and in addition

πn πn 1 l (R) 1 l l l (R) n > R R n | − | − − − p βR Cd (R) C e− . (6.12) > − βR Here we have used (5.17). Thus,e we have proved that outside an (e− )- neighborhood of the set , we have the estimate O Z πn hn(R) h( R ) 1 βR 6 CD− (R)e− (6.13) 1 ln(R) − 1 ln(R) − −b for R large enough. Using (6.13) after summation over n such that n MR/π | | 6 we get the assertion of the lemma. 2

M The next step is to study the asymptotic behavior of the integral sums IR for R . The next section is devoted to this subject. →∞ 7. Riemannian Integral Sums for Divergent Integrals Throughout this section we study the asymptotic behavior of integral sums with respect to small partitions of the interval of integration for functions having poles k of the form 1/(x a) , k > 1. Let φ(x) be a− smooth function on the interval [ M,M] outside some neigh- − borhood of a ®nite number of singular points a1,...,am ( M,M),where ∈ − φ(x)has poles of multiplicity deg ak,k = 1,...,m. More speci®cally, let us suppose that one can ®nd a rational function qφ such that the function φ qφ is smooth on [ M,M]. − For ®xed−ε>0, let us consider the Riemannian integral sum for the integral M φ(x) eipx dx, (7.1) M Z− which corresponds to the partition of the interval [ M,M] by the points − nε n: nε M { } | |6 ipx inεp Σε(φ e )=ε φ(nε) e . (7.2) n (M/ε) | |6X We will study the asymptotic behavior of such sums as ε 0 in the weighted → spaces L2,γ([ (π/ε),π/ε]),γ<0. We recall that these spaces denote L2-spaces − (γ/2) x with the exponential weight e | |.

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It is natural to divide the integral sum (7.2) into two parts. The ®rst part corresponds to the integral of a smooth function (some regularization of the integral (7.2))

M ipx (φ(x) qφ(x)) e dx (7.3) M − Z− and the second one is just the integral sum for the divergent integral

M ipx qφ(x) e dx. (7.4) M Z−

LEMMA 7.1. Let ψ be a smooth function. Then in the space L2,γ([ (π/ε),π/ε]), γ<0the following asymptotic representation is valid −

M ipx ipx Σε(ψ e )= ψ(x)e dx + M ψ C1([ M,M]) (ε). (7.5) M || || − O Z− Proof. For the smooth function ψ we have the estimate

M ipx ipx ipx Σε(ψ e ) ψ(x) e dx 6 εM sup (ψ(x)e )0 − M x [ M,M] | | Z− ∈ −

2 1/2 6 εM(1 + p ) ψ C1([ M,M]). (7.6) || || − The assertion of the lemma is a consequence of (7.6) and the inequality 2 1/2 2 1/2

π π 2 (1 + p ) L2,γ ([ , ]) 6 (1 + p ) L2,γ (R) < . (7.7) || || − ε ε || || ∞ Let us study the asymptotic behavior of the integral sums corresponding to the integral (7.4) in the limit ε 0. These integral sums appear to be partial Fourier series which fortunately→ can be computed explicitly. Let, say, the function qφ have a single simple pole at the point a 1 qφ = ,a(M,M). (7.8) x a ∈− − Then eipx einεp Σ = ε . (7.9) ε x a nε a ! n (M/ε) − | |6X −

LEMMA 7.2. In the space L2([ (π/ε),π/ε]) the following representation is valid (uniformly in ε, ε 0) − → inεx e πa iax (1/2) ε = π i sign x cot e + (M − ). (7.10) nε a − ε O n (M/ε) | |6X −

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Proof. First of all let us remark that the series einεx ε . nε a n Z X∈ − converges in the space L2([ (π/ε),π/ε]), when a/ε / Z. Moreover, we have the estimate − ∈

einεx ε nε a n >(M/ε) π π | | X − L2([ ε , ε ]) −

2π (1/2) 6 ε = (M − ). (7.11) v ε(εn a)2 O u n >(M/ε) − u| | X t It remains to verify that this series is the Fourier series on the interval [ (π/ε), π/ε] for the function π(i sign x cot(πa/ε)) eiax. Computing the Fourier− series for the functions eiωx and sign x−eiωx sin πω ( 1)n+1 eiωx = − einx (7.12) π n n ω X − and cos πω ( 1)n+1 1 einx i sign x eiωx = − einx + (7.13) π n n ω π n n ω X − X − and multiplying (7.12) by cot πω and subtracting the result from (7.13), we get einx = π(i sign x cot πω)eiωx. (7.14) n n ω − X −

Setting ω = a/ε after the change x xε, we get (7.10). 2 → Now it is easy to compute the integral sums corresponding to a pole of arbitrary multiplicity of the function qφ.

LEMMA 7.3. In the space L2([ (π/ε),π/ε]) for ε 0 the following representation is valid (uniformly in ε, ε− 0) → → einεp ε (nε a)m n (M/ε) | |6X − 1 dm 1 πa = − π i sign p cot eipa + (m 1)! dam 1 − ε − − m+(1/2) + (M − ). (7.15) O

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Proof. The fact that the in®nite series corresponding to the left-hand side of (7.15) converges to the function from the right-hand side of (7.15) can be heuristically obtained by differentiating (7.14) with respect to the parameter ω. The proof of the validity of such an operation can easily be given by a direct calculation of these series. The bound on the remainder is a consequence of the following estimates einεp ε (nε a)m n >(M/ε) π π | | X − L2([ ε , ε ]) −

2 π 6 ε v ε(εn a)2m u n >(M/ε) − u| | X t 1/2 dx m+(1/2) 6 2π 2m = (M − ). 2 (7.16) x >M (x a) ! O Z| | − Remark 7.1. We have thus found that every pole of multiplicity m of the m 1 iap function qφ at the point a gives rise to terms of the form P − (p, ε) e in m 1 the asymptotic representation (7.15), where P − (p, ε) is a polynomial in p of degree m 1 with coef®cients depending on ε − m 1 m 1 − k P − (p, ε)= ck(ε)p . (7.17) kX=0 These coef®cients have singularities at the points πak/ε Z and for ε 0 admit the estimates ∈ → πa k m c (ε)= εk+1 mdist ,Z − , (7.18) k − ε O ! where dist(x, Z) denotes the distance from the point x to the set Z. Collecting the results we obtained we can formulate the central result of this section as follows. THEOREM 7.1. Let φ some function on the interval [ M,M] such that there is a rational function − n Mk bm q (x)= k (7.19) φ (x a )m m= k kX=1 X1 − a1,...,am ( M,M) with the property that the function φ qφ belongs to the 1 ∈ − − class C ([ M,M]). Then in the space L2,γ([ (π/ε),π/ε]), γ<0, we have − −

inε M (1/2) lim sup ε φ(nε) e · Φε ( ) = (M − ), (7.20) ε 0 − · O → n 6(M/ε) π π | | X L2,γ ([ ε , ε ]) −

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.50 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 163 where M M ipx Φε (p)= (φ(x) qφ(x)) e dx + M − Z− n Mk m 1 m 1 d − + bk m 1 (m 1)! da − × kX=1 mX=1 −

πa ipa

π i sign p cot e . 2 (7.21) × − ε a=ak

8. Asymptotics of the Solution of the Integral Equation on a Finite Interval The results of Sections 5±7 allow us to study the asymptotic behavior of the solution fR (5.25) of the integral equation (5.11) for R in the weighted →∞ spaces L2,γ([ R, R]), γ<0, outside some neighborhood of the special set Z given by (5.18).− We recall that this set consists of those values of the parameter R for which the operator (1 LR) is not invertible. Let us recall that the set −Z is contained in some neighborhood of the ®nite m e union of lattices = k=1 n Z πn/ak . It is useful to specify this neighborhood more precisely.Z ∪ ∪ ∈ { } For κ>0 let us consider the set m πn κm πn κm Ξκ = e− , + e− . (8.1) a − a k=1 n Z k k [ [∈ In accordance with Lemma 5.2, for N large enough and κ<βwe have the inclusion

ZN Ξκ. (8.2) ⊂ We will show that under the condition that the right-hand side h of the integral equation (5.11) has a compact support, the asymptotics of the solution fR of (5.1) for R in the space L2,γ([ R, R]) can be studied for all κ satisfying the condition→∞κ<β/2p,wherepis− the maximum value of multiplicity of the roots of the equation l(s)=1. First let us note that Fourier transforms of functions h with a compact support are smooth. Let M be such a value that the interval [ M,M] contains all the − real roots a1,...,am, of the equation l(s)=1. Hence, there is only one rational function qh of the form

m Mk l bk qh(s)= l, (8.3) (x ak) kX=1Xl=1 −

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.51 164 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV where Mk = deg ak is the multiplicity of the root ak,k = 1,...,m, which is such that the function H(s) de®ned by the equation

h(s) H(s)= qh(s) (8.4) 1 l(s)− −b is a smooth function on the real axis (in particular it is smooth on the interval [ M,M]). We introduce a function Φh which is important for later use, since it − determines the asymptotic behavior of the solution fR of (5.1) as R : →∞ 1 Φ (p, R)= H(s)eisp ds + S (p, R), (8.5) h 2π h ZR where

Sh(p, R) m Mk l l 1 1 bk d − ipa = l 1 π(i sign p cot Ra) e a=ak . (8.6) 2π (l 1)! da − { − }| kX=1 Xl=1 − Theorem 7.1 provides necessary bounds to estimate the difference between the M integral sums IR from (6.4) for the integral (6.5) and the function Φh( ,R)given by (8.5). ·

LEMMA 8.1. Let h have compact support. For R outside the set Z we have the estimate →∞ ∪Z

M lim sup IR Φh( ,R) L2,γ ([ R,R]) 6 η(M), (8.7) R ,R/Z || − · || − →∞ ∈ ∪Z where η(M) is some function with the property

lim η(M)=0. (8.8) M →∞ Proof. From Theorem 7.1 it follows that for η(M) we can choose an appropriate estimate for the integral

is H(s) e · ds s >M Z| | L2,γ ([ R,R]) −

is 6 H(s) e · ds δ(M) (8.9) s >M ≡ Z| | L2(R)

plus terms of the order (M (1/2)): η(M)=δ(M)+ (M (1/2)).SinceH(s) O − O − belongs to L2(R), we conclude that property (8.8) holds. 2

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THEOREM 8.1. Let h have compact support. Under the condition β κ< (8.10) 2p the function Φh( ,R)de®ned by (8.5) and (8.6) is the leading term of the asymp- · totics of the solution fR in the space L2,γ([ R, R]),γ < 0 for R outside − →∞ the set Ξκ in the sense that

lim fR Φh( ,R) L2,γ ([ R,R]) = 0. (8.11) R ,R/Ξκ || − · || − →∞ ∈

Proof. Recall that for a given M, we have represented our solution fR as a M ®nite sum (6.2) containing 2[MR/π]+1 terms (we have denoted it by fR )and tails the `tails' fR given by (6.3). Lemma 6.1 states that there is such a function ζ(M) with the property limM ζ(M)=0 such that →∞ tails lim sup fR L2([ R,R]) 6 ζ(M). (8.12) R ,R/Z || || − →∞ ∈ ∪Z M Lemma 6.2 gives a bound on the difference between fR and the integral sum M IR de®ned in (6.4) M M √ 1 βR fR IR L2([ R,R]) 6 C(h) MD− (R)e− . (8.13) || − || − Let us point out that for R/Ξκ,Ris large enough, the function d(R) de®ned by (6.7) admits the bound from∈ below

1 κR d(R) > CR− e− . (8.14)

In fact by de®nition (8.1) of the set Ξκ for all n Z and k = 1,...,m,forR large enough, we have the inequality ∈ πn R C e κR, (8.15) a > − − k

where C is a positive constant. This proves (8.14). Therefore using the de®nition (6.9) of the function D(R) and (8.13) for such R, we conclude that M M √ 2p (β 2κp)R fR IR L2([ R,R]) 6 C(h) MR e− − . (8.16) || − || − Under condition (8.10), the right-hand side of (8.16) tends to zero when R →∞ outside the set Ξκ. Taking into account (8.5), (8.12), (8.16) and the fact that the natural embedding of the space L2,γ into the space L2 is bounded for γ<0, we immediately conclude that

lim sup fR Φh( ,R) L2,γ ([ R,R]) 6 ζ(M)+η(M). (8.17) R ,R/Ξκ || − · || − →∞ ∈

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Since the upper limit in (8.17) does not depend on M, passing to the limit

M , we get the assertion of the theorem. 2 →∞

9. The Composed Integral Equation on the Axis. Operator Approach

Let us consider in the space L2(R) the unbounded operator

R = W (1 R)W (9.1) B −L on the domain

2 ( R)= (W ), (9.2) D B D (α/2) x where W is the multiplication operator by the function e | | and (α/2) x Wf(x)=e | |f(x). (9.3)

2 Since the symmetric operator W RW on the domain (W ) admits a bounded L D extension to the whole Hilbert space L2(R) (in fact this extension is a compact operator), the operator R is self-adjoint on the domain (9.2), since it is a bounded perturbation of the self-adjointB operator W 2. Let R be such a value that the operator (1 R) has a bounded inverse, i.e. −L R/Z, then the point zero belongs to the resolvent set of R and ∈ B 1 1 1 1 − = W − (1 R)− W− . (9.4) BR −L 2 Since the operator R is a compact perturbation of the operator W , its essential spectrum coincidesB with the interval [1, ) ∞ σess( R)=[1, ) B ∞ and outside the interval [1, ) the spectrum of R is discrete with possible accumulation point 1. ∞ B On the space L2(R), let us also consider the symmetric operator (the `sandwiched' integral operator (1 )) A −L = W (1 )W (9.5) A −L de®ned on the domain

( )= (W2). (9.6) D A D Recall that the kernel L of the operator satis®es Condition (5.2). Under the 3 L condition β>2α, we can apply the abstract theory of the ®rst sections. This theory shows that when the equation l(s)=1 (9.7)

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.54 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 167 has solutions in the strip Πα/2 the operator is not self-adjoint but only symmetric and the information about its self-adjointA extensions can be found in Section 3. In Section 5 we have learned that the asymptotic behavior of the solution fR of the integral equation on the axis

(1 R)fR =h (9.8) −L for R , is described by the function Φh( ,R) given by (8.5) and (8.6). The →∞ · following assertion allows us to look at the function Φh( ,R) from the operator point of view. ·

LEMMA 9.1. Let h (W),R / Z,then ∈D ∈ Ψh(,R) ( ∗) (9.9) · ∈D A and

∗Ψh( ,R)=Wh, (9.10) A · where

(α/2) p Ψh(p, R)=e− | |Φh(p, R). (9.11)

Proof. We shall prove the lemma passing to the Fourier transform. All the objects in the Fourier representation shall be marked by the symbol .Letus come back to formulae (8.5) and (8.6). In accordance with the description of the domain of the adjoint operator ( ∗) (Theorem 1.1), we conclude thatb D A (α/2) p e Sh(p, R) ( ∗). (9.12) − | | ∈D A Since h d(W), the functionb h admits an analytic continuation into the strip ∈D Πα/2. Therefore there is a rational function qh of the form (8.3) such that the function b h(s) H(s)= qh(s) 1 l(s)− −b admits a meromorphic continuation into the strip Πα/2. Hence, there is only one rational function ph(s) having singularities only at the complex roots of the Equation (9.7) such that

H(s) ph(s) (W), (9.13) − ∈D (it is clear that ph is a regularc function on the real axis). By de®nition of ph,we have that

1 W − ph ( ∗). (9.14) ∈D A c b

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Moreover

(α/2) p 1 isp e− | | (H(s) ph(s)) e 2π − Z 1 d 2 = W − (H ph) (W )= ( ). (9.15) − ∈D D A Assertion (9.9) is now a consequencec of (9.14)b and (9.15). Assertion (9.10) can be proven by using the standard tricks of Section 3

1 1 (α/2) p ∗Ψh( ,R)= W−(H ph)+ ∗(W− ph +e Sh(p, R)) A · A − A − | | = W h W (1 l)(qh + ph)+W(1 l)phd+W(1 l)qh b b bc− − b c − −

= Wcbh. c c c2 (9.16)

From now on we assumecb that all real roots a1,...,am, of Equation (9.7) are simple. As we know from the general extensions theory, the set of all self-adjoint extensions of the symmetric operator with de®ciency indices (n, n) is in one- to-one correspondence with the groupA U(n) of all n n-unitary matrices. In our case n coincides with the total number of roots of× (9.7) counting multiplicity in the strip Πα/2. In order to study the asymptotic behavior of the resolvents of the operators R for R , we need a special subfamily of the family of all self-adjoint extensionsB of→∞parametrized by elements of an m-dimensional torus Tm naturally embeddedA into the group U(n),wheremis the number of real roots of (9.7). Let us describe such a subfamily in the Fourier representation. As in Section 1, let be the space of rational functions (vanishing at the in®nity) having poles Q only in the strip Πα/2, such that the function (1 l(s))q(s),q ,is an analytic function in this strip. The space can be represented− as a sum∈Q of two subspaces Q = reg + sing, (9.17) Q Q Q where reg is the subspace of functions from being regular on the real axis Q Q and sing is the linear subspace generated by the simple fractions Q 1 ,k=1,...,m. (9.18) s ak − iα iαm m For given ω =(e 1 ,...,e ) T , let us de®ne a linear subspace ω ∈ D ⊂ L2(R) in the following manner + ω = U (α/2)q + Uα/2q−,q reg + D { − ∈Q }

+e U (α/2)q1 +eUα/2q2,q1,q2 sing , (9.19) { − ∈Q } + where q and q−e are de®nede by (2.26) and (2.27) provided that

iαk Res s=a q1(s)=e Res s=a q2(s),k=1,...,m. (9.20) | k | k

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LEMMA 9.2. The restriction ω of the adjoint operator to the domain A A∗

( ω)= ( )+ ω b b (9.21) D A D A D is a self-adjointb operator.b +,k ,k Proof.Letf ω and xa and xa− be coordinates of the element f with respect to the basis∈D (3.24), (3.25). As in (3.54), we have

+,k xa = 0, for Im a<0, (9.22)

,k xa− = 0, for Im a>0. (9.23)

The coordinates x corresponding to the real roots a ,...,a of the Equation a±k 1 m (9.7) are now connected by the relations

x+ = iαk x . ak e a−k (9.24) In the coordinates corresponding to the canonical basis (3.49), (3.50), relations (9.22), (9.23) can be rewritten as

+,k ,k ξ = iξ− , for Im a>0, (9.25) a − a¯

+,k ,k ξa = iξa¯− , for Im a>0 (9.26) and (9.24) transforms to the relation

ξ+ = cot α ξ , for a>0,k=1,...,m. (9.27) ak k a−k Equations (9.25)±(9.27) de®ne some self-adjoint relation in C and therefore, in

accordance with the theory of Section 3, the operator ω is self-adjoint. 2 A b LEMMA 9.3. Let R ω(R) be the trajectory on the torus Tm →

2ia1R 2iamR ω(R)= (e− ,...,e− ) (9.28) − and R does not belong to the exceptional set . Then, for every h (W),we have Z ∈D

Ψh( ,R) ( ) (9.29) · ∈D Aω(R) and

Ψh( ,R)=Wh. (9.30) Aω(R) ·

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Proof. In the case of simple roots using (8.5), (8.6) and (9.11), we have the following representation for the function Ψh(p, R):

Ψh(p, R) 1 = e (α/2) p H(s) eisp ds + 2π − | | Z 1 m iakp (α/2) p + Res qh(s) s=a π(i sign p cot akR) e − | |. (9.31) 2π | k − kX=1 To prove (9.29) let us compute the coordinates x with respect to the basis a±k (3.24), (3.25) of the part of Ψh( ,R)from the de®ciency subspace ( ∗) ( ), which coincides with the second· term in (9.31). Passing to the FourierD A transform\D A in (9.31), we see that

+ 1 + i cot akR x = Res qh(s) s=a (9.32) ak 2 | k and

1 i cot akR x− = − Res qh(s) s=a . (9.33) ak 2 | k Therefore we have the relation

+ 2ia R x = e− k x− ,k=1,...,m, (9.34) ak − ak

which proves (9.29). Now (9.30) is a consequence of Lemma 9.1. 2

CONSEQUENCE 9.1. For R/ the kernel of the operator ω(R) is empty . Hence, has a bounded inverse∈Z and A Aω(R) 1 − h =Ψ 1 (,R), (9.35) Aω(R) W− h · i.e. 1 − h(p) Aω(R) 1 m 1 1 (α/2) p W − h(s) W − h(ak) ips = e− | | + e ds 2π 1 l(s) l0(ak)(s ak) ! − Z −d kX=1 d− 1 m W 1h(a ) − k iakp (α/2) p (i sign p cot akR) e − | |. (9.35a) −2 l0(ak) − kX=1 d Proof. If the parameter R does not belong to the ®nite union of lattices , Z then for every k, k = 1,...,m, we infer that akR/Zand therefore all the components of the vector ω(R) are not equal to the value∈ 1. Hence, by Lemma −

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2.1, we conclude that Ker ω(R) = . Using now Remark 4.3, we see that 0 belongs to the resolvent setA of ,i.e.∅ has a bounded inverse. In this Aω(R) Aω(R) situation, (9.35) is just a consequence of (9.30). 2

LEMMA 9.5. Let h have a compact support and β κ< . (9.36) 2

For R outside the set Ξκ we have →∞ 1 1 lim PR( ω−(R) R− )h=0, (9.37) R ,R/Ξκ A −B →∞ ∈ where PR is the projector onto the subspace L2(R [ R, R]). \ − Proof.ForRlarge enough outside the set Ξκ the operator (1 R)can be −L inverted and therefore the operator R has a bounded inverse B 1 1 1 1 − = W − (1 R)− W− . (9.38) BR −L Moreover, since Ξκ, by Consequence 9.1 we conclude that ω(R) also has a bounded inverseZ⊂ and (9.35) holds. Under condition (9.36), theA assumptions of Theorem 8.1 are satis®ed for every γ<0. Therefore, for the particular case γ = α, we have the asymptotic representation in the space L2, α(R) − − 1 1 (1 R)− W− h Φ 1 (,R)=o(1), (9.39) −L − W− h · for R outside the set Ξκ. By de®nition of the space L2, α(R), we see that →∞ − the convergence of a sequence fn(p) in the space L2, α(R) is equivalent to { (α/} 2) p − the convergence of the sequence e− | |fn(p) in the space L2(R).Taking into account this remark and noting{ that } (α/2) p 1 1 1 1 1 1 e− | |(1 R)− W− h=W− (1 R)− W− = − h (9.40) −L −L BR and (α/2) p 1 e− | |Φ 1 ( ,R)=Ψ 1 (,R)= − h, (9.41) W − h · W− h · Aω(R)

we get (9.37) as a consequence of (9.39). 2

Lemma 9.5 gives a result concerning the asymptotic behavior of the family 1 1 of operators PR( − B− ) as R outside the set Ξκ. One can show Aω(R) − R →∞ that the result corresponding to (9.37) without the projector PR still holds only for κ<α/2 but it does not hold when R Ξα/2 Ξβ/2 and R (we recall that we have assumed the inequality β>∈ 3α/2\ from the very→∞ beginning of this section). We need, however, some results concerning the asymptotics on β R the whole space L2(R) outside the (e− 2 )-neighborhood of the exceptional O 1 set . In order to obtain such a result, we note that the operator − is some Z Aω(R)

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®nite-rank perturbation R of the resolvent of the special self-adjoint extension K ω of the operator corresponding to the point A ∗ A m ω∗ =(1,...,1) T . ⊂ In fact, we have here a particular case of Krein's formula

1 1 − = − + R, (9.42) Aω(R) Aω∗ K where R is the ®nite-rank operator K m R = µk(R)( ,hk)hk. (9.43) K · · kX=1 Here hk,k =1,...,m, are elements of L2(R) of the form

iakp (α/2) p hk(p)=e − | | (9.44) and 1 µk(R)= cot akR, (9.45) 2l0(ak) where l0(ak) are the values of the ®rst derivative of the function l.

CONSEQUENCE 9.2. Let h have compact support and κ<β/2. For R →∞ outside the set Ξκ we have

1 1 lim ( − + − )h=0, (9.46) ω∗ R R R ,R/ Ξκ A K −B →∞ ∈ e where R is the ®nite-rank operator K eR = PR RPR. (9.47) K K

Proof.e Let us write the explicit expression for the resolvent of ω∗ using (9.42) and (9.35a) A

α p 1 m 1 1 e−2|| W−h(s) W−h(ak) isp − h(p)= + e ds ω∗ A 2π 1 l(s) l0(ak)(s ak) ! − Z −d kX=1 d− m 1 i W − h(ak) ia p (α/2) p sign p e k − | |. (9.48) −2 l0(ak) kX=1 d Now it easy to see that

1 (I PR) ω− h R 0. (9.49) − A ∗ −→ →∞

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1 In addition, since the operator R− acts on the subspace L2(R [ R, R]) simply 2 B \ − as W − , we infer that 1 2 (1 PR) R− =(1 PR)W− R 0. (9.50) − B − −→ →∞ Thus assertion (9.46) is a consequence of (9.37), Krein's formula (9.42) and

Equations (9.49)±(9.50). 2

10. The Asymptotic Behavior of the Regularized Operator in the Strong Resolvent Sense In this section, we obtain the central result of the paper concerning the asymptotic behavior of the resolvents of R for R B →∞ Rz( R) R Rz( ω(R)), (10.1) B ∼ →∞ A where ω(R) is the trajectory on the torus Tm, ω(R)= (e 2ia1R,...,e 2iamR) − − − and a1,...,am are real roots of the equation l(s)=1andandbyRz(A)we denote the resolvent of the operator A at the point z. Consequence 9.2 is the key result which allows us to prove the asymptotic β R (10.1) for R outside the (e− 2 )-neighborhood of the exceptional set . In order to obtain→∞ the correspondingO result in this neighborhood, we use the factZ that the operator-valued function ω Rz( ω(R)) is uniformly continuous on the torus Tm and some rough arguments7→ of perturbationA theory, which allows us to control the ¯uctuations of the function Rz( R). Under the condition β>4α, these arguments extend the result to the wholeB real axis. In order to realize this program, we need to study some intermediate object which, on the one hand, gives a good description of the asymptotics Rz( R) for B R and, on the other hand, is a small perturbation of Rz( ω(R)). →∞ m m A For given Λ=(λ1,...,λm) R = R , let us consider the ®nite- rank operator ∈ ∗ ∪{∞}

m Tz(Λ,R)= Ωik(Λ,R;z) ik(R), (10.2) Q i,kX=1 where

ik(R) =( , ω PRhi) ω PRhk (10.3) Q · · A ∗ A ∗ are rank-one operators, and the matrix Ω(Λ,R;z) is de®ned by

1 Ω(Λ,R;z)=(Λ+E(R, z))− . (10.4) Here by Λ we denote the diagonal matrix

Λ=diag(λ1,...,λm) (10.5)

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.61 174 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV and the matrix E(R, z) is given by its matrix elements 1 Eik(R; z)=(( ω z)− ω PRhk, ω PRhi). (10.6) A ∗ − A ∗ A ∗ In the case where some elements of the the matrix Λ are equal to in®nity, we simply omit the corresponding terms in (10.2). Let us note that the elements PRhk 2 obviously belong to the domain ( ω∗ ),sincePRhk (W) ( ω∗). Therefore, the matrix E(R; z) is wellD A de®ned. We also have∈D to prove that⊂D detA(Λ+ E(R; z)) = 0. 6 LEMMA 10.1. For R large enough the matrix Λ+E(R;z) has an inverse for all Λ Rm and z C R. Proof∈ .∗ First of all∈ let\ us note that the limit (z) lim E(R; z) AR , (10.7) E ≡ R − →∞ R R with A the matrix Aik =( ω∗PRhk,PRhi) exists. The proof of this assertion is based on the representationA

Eik(R; z)=( ω PRhk,PRhi)+z(PRhk,PRhi)+ A∗ 2 1 +z (( ω z)− PRhk,PRhi). (10.8) A ∗ − The last two terms in (10.8) have a limit, since we have the convergence PRhk → hk,k =1,...,m,in the space L2(R). To compute the ®rst term we stress that the elements PRhk belong to the domain of the symmetric operator = W (1 )W and therefore A −L

( ω PRhk,PRhj) A ∗ =(W(1 )WPRhk,PRhj) −L iak iaj =((1 )WPRhk,WPRhj)=((1 )e ·, e ·) −L −L iak iaj =(e ·, e ·)L2([ R,R]) e − −

ia x iaj y dx dyL(x y)e k − . (10.9) − [ R,R] [ R,R] − ZZ− × − iax+iby Let us compute the double integral of the function f(x, y)=L(x y)e− on the square [ R, R] [ R, R]. Using the representation − − × − dx dyf(x, y) [ R,R] [ R,R] ZZ− × − √2R ξ+√2R ξ + η η ξ = − dξ − dηf , − + 0 ξ √2R √2 √2 Z Z − 0 ξ+√2R ξ + η η ξ + dξ dηf , − (10.10) √2R ξ √2R √2 √2 Z− Z− −

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.62 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 175 we immediately get that for a = b

dx dyf(x, y) [ R,R] [ R,R] ZZ− × − R R iax iax = 2R L(x) e− dx L(x) e− x dx + R − 0 Z− Z 0 iax + L(x) e− x dx (10.11) R Z− and for a = b 6 dx dyf(x, y) [ R,R] [ R,R] ZZ− × − R 1 i(b a)R ibx = e − dxL(x) e− i(b a) 0 − − ( Z 0 R i(b a)R ibx i(b a)R iax e− − dxL(x) e− e− − dxL(x) e− + − R − 0 Z− Z 0 +i(b a)R iax + e − dxL(x)e− . (10.12) R ) Z− Taking into account that the Fourier transform of the kernel L at the points a1,...,am, takes the value 1 and computing explicitly the inner products ia ia (e ·, e ·)L2([ R,R]) = 2R − and

ia ib sin(b a)R (e ·, e ·)L2([ R,R]) = 2 − , − b a − we infer that

lim ( ω PRhk,PRhj)< . (10.13) R A ∗ ∞ →∞ Therefore, the limit matrix (z) from (10.7) has the form E (z)=zG + z2F(z), (10.14) E where G is nothing but the Gram matrix of the system of vectors hk { } Gik =(hk,hi) (10.15) and

1 Fik(z)=(( ω z)− hk,hi). (10.16) A ∗ −

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Let be an arbitrary self-adjoint matrix. We will prove that the matrix + zG + zF2F(z) has an inverse for all z, Im z = 0. In fact the function F 6 z + zG + z2F(z) M(z) (10.17) 7→ F ≡ 1 is a matrix-valued R-function in the upper-half plane, i.e. i− [M(z) M ∗(z)] is a positive de®nite matrix. From the general theory of R-functions (see,− e.g., [43]) we know that if some R-function has the inverse at some point z of the complex plane, then that remains true for all z C R. But for values of z which are small enough, the fact that the matrix from∈ (10.17)\ has an inverse is a consequence of the following two assertions: (i) the matrix G is positive de®nite as every Gram matrix of a linearly independent system of vectors and (ii) z2F (z)= (z2), O since the operator ω has a bounded inverse and, therefore, A ∗ 1 lim(( ω z)− hk,hi)< . z 0 A ∗ − ∞ → At least for z small enough we also have the uniform estimate 2 1 1 ( + zG z F(z))− C Im z − , (10.18) || F − || 6 | | the constant C being independent on the matrix . In fact, this is a consequence of the equality F 2 1 1 2 1 1 ( + zG + z F(z))− =(1+( +zG)− z F(z))− ( + zG)− (10.19) F F F and the estimate 1 (1/2) (1/2) (1/2) 1 (1/2) ( + zG)− = G− (G− G− + z)− G− || F || || F || (1/2) 2 (1/2) (1/2) 1 G− (G− G− + z)− 6 || || || F || (1/2) 2 1 G− Im z − . (10.20) 6 || || | | Here we used the facts that the matrix G is self-adjoint, it has an inverse and the (1/2) (1/2) matrix G− G− is also self-adjoint. Hence, we haveF proved that for R large enough the matrix-valued function Tz(Λ,R) is well-de®ned for all values of Λ including the in®nite ones. More- over, from (10.18) we conclude that the matrix elements Ωik(Λ,R;z) admit the estimate

Ωik(Λ,R;z) C(z), (10.21) | | 6

which holds uniformly in Λ and R for R large enough (when z is small enough). 2

Now we are in a position to introduce, for all Λ Rm and R large enough, the operator-valued function ∈ ∗

z(Λ,R)=Rz( ω ) Rz( ω )Tz(Λ,R)Rz( ω ). (10.22) R A ∗ − A ∗ A ∗ Simple computations show that the following assertion holds:

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1 LEMMA 10.2. Let the operator − + R have an inverse denoted by BR Aω∗ K 1 1 BR =( − + R)− . e (10.23) Aω∗ K Then the resolvent ofeBR has the form

Rz(BR)= z(Λ(R),R), (10.24) R where

Λ(R)=2(l0(a1)tan a1R,...,l0(am)tan amR). (10.25)

Using now Consequence 9.2, we have an important result concerning the κR asymptotic behavior of the resolvents Rz( R) for R outside an (e− )- neighborhood of the exceptional set . B →∞ O Z THEOREM 10.1. Let h have a compact support and κ<β/2.Then

lim (Rz( R) z(Λ(R),R))h = 0, (10.26) R ,R/Ξκ B −R →∞ ∈ where z(Λ,R) is the operator-valued function given by (10.22) and Λ(R) is de®nedR by (10.25). Proof. We shall systematically use the equality

1 1 (A z)− (B z)− − − 1 −1 1 1 1 =(1 zB− )− (1 + z(A z)− )(A− B− ), (10.27) − − − which holds for arbitrary self-adjoint operators A and B such that 0 ρ(A) ρ(B). Using (10.27), we get ∈ ∩

1 1 Rz(A) Rz(B) C(z) A− B− , (10.28) || − || 6 || − || where C(z) does not depend on A and B. 1 When the operator − + has an inverse, by (10.28) we immediately get ω∗ R that A K e 1 1 Rz(BR) z(Λ(R),R)h 6 C(z) ( ω− + R − )h . (10.29) || −R || || A ∗ K −BR || 1 This estimate remains true also in the case where thee operator − + does ω∗ R not have an inverse, as seen by simple perturbation arguments.A We recallK that e m R = µk(R)( ,PRhk)PRhk, (10.30) K · · kX=1 e where µk(R) are de®ned by (9.44). By small perturbations of the coef®cients 1 µ , we can always make the operator − + R invertible and therefore after k ω∗ such a perturbation, we get the analogA of theK estimate (10.29). But the small e

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.65 178 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV perturbations of µk give rise to small perturbations (with respect to the topology of the compacti®cation Rm) of the components of the vector Λ. On the other ∗ hand, we have proved that the function z(Λ,R) is a continuous function in Λ Rm. From this, the estimate (10.29)R can be proved by a limit procedure. Now∈ the∗ required assertion (10.26) of the lemma follows from (10.29) and the

result of Consequence 9.2. 2

m THEOREM 10.2. For given Λ R the operator-valued function z(Λ,R)has a limit, as R , in the topology∈ ∗ of norm-convergence R →∞ n lim z(Λ,R)=Rz( ω(Λ)), (10.31) − R R A →∞ where 2l (a ) ω(Λ) = exp 2i arctan 0 1 ,..., λ − − 1 2l (a ) exp 2i arctan 0 m (10.32) λ − m ! under the natural de®nition 2l (a) exp 2i arctan 0 = 1 (10.33) λ − − for λ = 0. Moreover, the convergence (10.31) holds uniformly on Λ Rm. Proof. First take Λ Rm and assume that all the components of∈ Λ∗ differ ∈ from zero. For R large enough, the function z(Λ,R) is correctly de®ned not only for complex values of z but also at theR point z = 0. This means that the 1 operator − + (Λ) has an inverse, where (Λ) is given by (10.30) for Aω∗ K K 2l0(ak) µk = e . e (10.34) λk Let us denote this inverse by B(Λ)

1 1 B(Λ) = ( − + (Λ))− . (10.35) Aω∗ K Then e 1 B(Λ)− = z(Λ,R) (10.36) R and therefore the function z(Λ,R) coincides with the resolvent of B(Λ) R z(Λ,R)=Rz(B(Λ)). (10.37) R But for a ®xed Λ, we obviously have 1 1 n lim ( ω− + (Λ)) = − . (10.38) − R A ∗ K Aω(Λ) →∞ e

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Again using inequality (10.28) we infer that

n lim z(Λ,R)=Rz( ω(Λ)). (10.39) − R R A →∞

Let us prove the existence of the limit z(Λ,R) for R in the case where some of the components of Λ may coincideR with zero or→∞ in®nity. Let us note that the function z(Λ,R) can also be represented in the form R

z(Λ,R)=Rz( ω ) Rz( ω ) ω Pz(Λ,R) ω Rz( ω ), (10.40) R A ∗ − A ∗ A ∗ A ∗ A ∗ where

m Pz(Λ,R)= Ωi,k(Λ,R;z) ik(R) (10.41) P Xi,k and

ik(R)=( ,PRhi)PRhk P · (if some components of Λ are equal to in®nity the corresponding terms in the sum (10.41) should be omitted). It is easy to understand that the operators ik(R) converge as R P →∞

lim ik(R)=( ,hi)hk. (10.42) R P · →∞

Taking into account this remark and the fact that the operator Rz( ω∗ ) ω∗ can naturally be extended to a bounded operator, we can prove by (10.40)A A the m convergence of z(Λ,R) for every Λ R . Using (10.21),R we conclude that the∈ convergence∗ (10.31) holds uniformly on Λ (with respect to the topology of the compacti®cation Rm). Therefore, the limit operator is a continuous function in Λ. On the other hand,∗ using standard arguments of the extension theory of symmetric operators, we see that the family of extensions is also a continuous function in Λ, which proves the last Aω(Λ) assertion of the theorem. 2

As a consequence of Theorems 10.1 and 10.2, we have the following result:

THEOREM 10.3. Let R outside the set Ξκ for κ<β/2, then we have the strong convergence →∞

lim Rz( R) Rz( ω(R)) h=0 (10.43) R ,R/Ξκ{ B − A } →∞ ∈ on a dense set of functions h having compact support.

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κR Proof. Theorem 10.1 shows that, for R , outside an (e− )-neighborhood of the exceptional set , we have the asymptotic→∞ representationO in the sense of the strong convergence onZ a dense set of functions with compact support

Rz( R) R z(Λ(R),R). (10.44) B ∼ →∞ R At the same time, Theorem 10.2 describes the limit behavior of the family z(Λ,R) as R for ®xed values of Λ. Using that the convergence in TheoremR 10.2 holds→∞ uniformly on Λ, we get the convergence on the concrete trajectory ω(R) in the sense of the norm convergence

z(Λ(R),R) R Rz( ω(R)). (10.45) R ∼ →∞ A So we have proved the convergence to zero of the family of uniformly bounded operators Rz( R) Rz( ) on a dense set. It is a well known fact that this B − Aω(R) implies strong convergence. 2

Thus, we have proved the result concerning the description of the asymp- κR totics of the operators R in the strong resolvent sense outside an (e− ) neighborhood of the exceptionalB set . Inside this neighborhood, weO use standard perturbation theory arguments. Z

LEMMA 10.3. Let R

Proof. Let us note that the difference R R B −B 0 R R =W( R R )W (10.47) B −B 0 L −L 0 admits a bounded extensione ande αR R R e R R . (10.48) ||B −B 0|| 6 ||L − L 0 || We recall that in accordancee withe (0.9a), R is an integral operator with the kernel L x y L (x, y)=χ Lreg(x y)χ , R R R R − reg where χ(x) is the characteristic function of the interval [ 1, 1] and LR is a periodic extension with period 2R of the function L from the− interval [ R, R] to − the whole axis. Let PR is the orthogonal projection from L(R) onto the subspace

L2([ R, R]). Then the difference ∆ = R R0 of integral operators can be represented− as L L − L e e ∆ = PR ∆ PR +∆P∆ PR +PR ∆ ∆P +∆P∆ ∆P, L 0 L 0 L 0 0 L L

ACAP1268.tex; 22/07/1997; 16:51; v.7; p.68 ATTRACTORS IN A MODEL RELATED TO THE THREE-BODY QUANTUM PROBLEM 181 where

∆P = PR PR . − 0 Since the function L is HolderÈ continuous by assumption, using the representation

(0.9a) for the kernels of the integral operators R, R0 , we immediately conclude that L L

PR ∆ PR = R (√∆R), ∆R 0. (10.49) || 0 L 0 || O → That gives the norm estimate for the ®rst term in (10.49). To estimate the remaining ones, we note that ∆P ∆ =∆P R and ∆ ∆P = R∆P . The operators L L L L R∆P and ∆P R are adjoint to each other and therefore have the same norm. L Hence, the normL of the sum of the last three terms in (10.49) can be estimated by 3 R∆P . Let us estimate the norm of R∆P .Letf L2(R),then ||L || L ∈ ( R∆P)f(x) |L | per = LR (x y)f(y) dy [ R,R] [ R ,R ] − Z− \ − 0 0

√ 6 C f(y) dy 6 C 2∆R f L2(R). (10.50) [ R,R] [ R ,R ] | | || || Z − \ − 0 0 Using this rough estimate, we have √ R∆Pf 6 C ∆R 1 L2([ R,R]) f (10.51) ||L || || || − || || and, therefore, 1/2 R∆P CR √∆R. (10.52) ||L || 6 Collecting (10.50) and (10.53) together, we conclude that ∆ CR (√∆R), ∆R 0 || L|| 6 O → for R large enough. Using now Hilbert's equality and (10.48), we get the assertion

of the lemma. 2

Now we are in a position to prove the main result of this paper.

THEOREM 10.4. Let a1,...,am be simple real roots of the equation l(s)=1 and ω(R) be the trajectory on the torus Tm given by

2ia1R 2iamR ω(R)= (e− ,...,e− ). (10.53) − Under the condition β>4α, (10.54) we have the following asymptotics in the strong resolvent sense

s lim Rz( R) Rz( ω(R)) = 0. (10.55) − R { B − A } →∞

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Proof. Under Condition (10.54), we can ®nd κ such that 2α<κ<β/2. Theorem 10.2 gives us the information needed to draw our conclusions outside an (e κR)-neighborhood of the exceptional set and Lemma 10.3 shows that, O − Z inside this neighborhood, the ¯uctuations of R are arbitrary small for R . B →∞ Taking into account that the operator-valued function Rz( ω(R)) is a uniformly continuous function in R, we get the assertion of the theoremA as a consequence

of Theorem 10.3 and Lemma 10.3. 2

Acknowledgements We are grateful to Professors L. Bogachev, J. Brasche, V. Buslaev, S. Cheremshan- tzev, G. F. Dell'Antonio, L. D. Faddeev, R. Figari, B. Pavlov, A. Ponosov, S. Scar- latti, and A. Teta for fruitful discussions. One of us (K.A.M.) would like to acknowledge the hospitality of the `FakultatÈ furÈ Mathematik' of Ruhr Univer- sitatÈ Bochum and also thank the Alexander von Humboldt Stiftung for ®nancial support.

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