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

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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. Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem 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 sys- tems 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 direc- tion 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 oper- ator 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 ZAL-ALD PIPS No.: 115457 MATHKAP ACAP1268.tex; 22/07/1997; 16:51; v.7; p.1 114 SERGIO ALBEVERIO AND KONSTANTIN A. MAKAROV 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 unbound- ed 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 `experi- mental' 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 interac- tions 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 coor- dinates, λα(ε) are coupling constants and ε 0 is a small scaling parameter.
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