is the so-called Efimov effect: if in a system of three particles interacting by means of ... In the present paper we consider a system of three identical quantum ...
On E f i m o v ' s Effect in a S y s t e m of T h r e e I d e n t i c a l Quantum Particles S. N . Lakaev
UDC 517.984 Introduction
One of the remarkable results of spectral analysis of the continuous three-particle Schrhdinger operator is the so-called Efimov effect: if in a system of three particles interacting by means of short-range pair potentials none of the three two-particle subsystems has bound states with negative energy but at least two of them are in resonance with energy at zero, then this three-particle system has infinitely many three-particle bound states with negative energy, and these states accumulate at zero. This effect was first discovered by Efimov [1]. Since then this problem has been investigated in many physical studies [2-4]. A rigorous mathematical proof of the existence of Efimov's effect was originally carried out in [5] and then in [6, 7]. Yafaev [5] proved the existence of Efimov's effect using the Faddeev integral equation method. On the other hand, Ovchinnlkov and Sigal [6] applied an interesting variational method to establish Efimov's effect for a system of three particles, two of which are heavy (2H) and one light (1L), under the assumption that only the H-L subsystems, interacting by means of spherically symmetric pair potentials, are in resonance with energy at zero. Furthermore, Tamura [7] used the variational method of [6] to prove the existence of Efimov's effect without constraints on the masses of particles interacting by means of (not necessarily spherically symmetric) pair potentials for the case in which all two-particle subsystems are in resonance with energy at zero. In models of solid physics [8] and also in the lattice field theory [9], the so-called discrete Schrhdinger operators are considered, which are lattice counterparts of the ordinary three-particle Schrhdinger operator in a continuous space. The presence of Efimov's effect for these operators can also be expected naturally because, as is shown by a thorough analysis of the effect [5], it is due to the "long-wave" part of the spectrum of the Schrhdinger operator, which is the same in both the lattice and continuous cases. At physical level this was proved in [8, 10]. In [11] the author stated a result on Efimov's effect for a system of three arbitrary quantum particles interacting by means of zero-range attractive pair potentials with fixed value of the total quasi-momentum of the system. Only a sketch of proof of the main result was given there. In the present paper we consider a system of three identical quantum particles (bosons) on a threedimensional lattice that interact by means of zero-range attractive pair potentials, and present a reasonably complete proof of the existence of Efimov's effect. Moreover, the dependence of this phenomenon on the quasi-momentum is analyzed. Specifically, we describe the set of values of the quasi-momentum for which Efimov's effect occurs. It should be noted that the study of this symmetric case is most important because Efimov's effect apparently does not exist in the antisymmetric case, and neither does it exist in the continuous antisymmetric case [12]. Although the energy operator of a system of three lattice particles is bounded and the perturbation in the pair problem is a compact operator, the character of Efimov's effect for a system of three quantum lattice particles is more complex than in the continuous case. In the latter case the energy of center-of-mass motion can be separated out from the total Hamiltonian so that the three-particle "bound states" are eigenvectors of the energy operator with total momentum separated (and this operator does not, in fact, depend on the values of the total momentum). Therefore, Efimov's effect either exists or does not exist for all values of the total momentum simultaneously. In lattice terms, the "center-of-mass separation" corresponds to a realization of the Hamiltonian as a "fibered operator", i.e., as the "direct integral of a family of operators" H(K) depending on the values of the total A. Navoi State University, Samarkand. Translated from Funktsionaltnyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 1528, July-September, 1993. Original article submitted November 22, 1991. 166
0016-2663/93/2703-0166512.50
(~)1993 Plenum Publishing Corporation
quasi-momentum K E T a (T a is the three-dimensional torus). In this case a "bound state" is an eigenvector of the operator H(K) for. some K ~ T 3 . Typically, this eigenvector varies continuously with K . Therefore, Efimov's effect exists only for the values of K lying on some manifold 9)I of codimension 1. This paper consists of five sections. In §1 the energy operators for two and three lattice particles are described as self-adjoint operators in a Hilbert space. The dependence of the spectrum of the two-particle operator on the quasi-momentum of these particles and on the coupling constant is studied in §2. In this section the main result is stated (Theorem 2.1). In §3 the study of the eigenvalues of the three-particle operator is reduced to the investigation of a Faddeev-type integral equation. And in §§4 and 5 the spectral properties of the Faddeev-type operator are considered and Theorem 2.1 is proved. The author expresses his warmest gratitude to R. A. Minlos, A. G. Kostyuchenko, and A. I. Mogil~ner for their attention and discussion of the results and also to the reviewer of the paper for valuable remarks. §1. E n e r g y O p e r a t o r s for T w o a n d T h r e e P a r t i c l e s Let Z 3 be a three-dimensional integral lattice, ( z a ) m = Z a x ... x Z a the ruth Cartesian power of Z a , /2((Z3) "~) the Hilbert space of square summable functions ~ on (Za) m , and /~((Za) "~) C / : ( ( Z a ) ~) the subspace of functions ~(n~, ... ,nm) symmetric with respect to the permutation of any two v~iables. In the coordinate representation the H ~ i l t o n i a n s of systems of two and three identical particles (bosons) on a three-dimensional lattice act in the spaces l~((Za) :) and l~((Za)a), respectively, according to the formulas
g(s)[~(ni + s, n~) + ~ ( n l , n~ + s)] - p 6 , ~ , ~ ( n l ,
~ , ~ ( n i , n2) = ~
n~),
(1.1)
sEZ ~
~ , ~ ( " 1 , ~2, "3) = ~ g ( $ ) [ 6 ( " 1 + $, ~2, ~3) + 6(~1 , "2 + 8, ~3) + ~(~1 , ~2, "3 + 8)] +
+
Here g(s), s = (sO), s(~), s(a)) e Na, is a real-valued functioa on ga depending only on ls(~)l, ~s(~)~, and ls(a)~ and satisfying the inequality li(~)l ~ C e x p { - a l ~ l } ,
~ = !¢~t + ~ ( ~ + ~ ¢ ~
for some a > 0 and C > 0; ~ > 0 is the interaction energy and ~mn the Kronecker del~a. Le~ T a be the three-dimensional torus, Le ((T a)m) the Hilbert space of square integrable functions on (Ta) ~ , and L~((Ta) m) C L~((Ta) ~) the subspace of functions ¢ ( k ~ , ... , k~) symmetric with respect to permutation of any two variables. In the momentum representation the Hamiltonians of systems of two and three identical particles act in the Hilbert spaces L~((Ta) ~) and L~((Ta)a), respectively, according to the formul~ -
3 H ~ ' ( ~ I ' ~2' ~3) = ( ~ (~~ a ) )3
.
~
-
-
I
t
I
I
~:) d~d~:
I
,
(~.a)
) ,`~1_ ' ~2'
a .[
.
.
~
k ~ 5~(k ~
,
,
' ' ', dk~dkzdk~,
(1.4)
~1 ~ O, and a(~) ~ a(z') + ~ . We now show that there are infinitely many points z < 0 such that 1 is an eigenvalue of T ( K , z). Lemma 4.4 implies that there exists a z' such that lIT(K, z)]l < i for z < z' and, consequently, ~ (z') = 0. As was shown above, for any N > 0 there exists z" < 0 and ~ > 1 such that w~(z") = N. Applying the proposition, we conclude that for some g < z" the operator T ( K , ~) has eigen~lue 1 of multiplicity ~ , and ~ (g) = ~. Furthermore, the proposition can be applied once again if we take z ~ = ~. By repeating this procedure su~ciently many times we show that there are at least N points z < 0, where the operator T ( K , z) has eigenvalue 1 (the points z are counted with multiplicities of the corresponding eigenvalues). ~
~
References
1. V. N. Efimov, "Bound states of three resonantly interacting particles," Yadernaya Fizika, 12, No. 5, 1080-1091 (1970). 2. R. D. Amado and J. V. Noble, "Efimov's effect: A new pathology of three-particle systems. I," Phys. Left. B, 35, No. 1, 25-27 (1971). 3. R. D. Amado and J. V. Noble, "Efimov's effect: A new pathology of three-particle systems. II," Phys. Rev. D, 5, No. 8, 1992-2002. 4. S. P. Merkurtev and L. D. Faddeev, Quantum Scattering Theory for Systems of Several Particles [in Russian], Nauka, Moscow (1985). 5. D. R. Yafaev, "On the theory of discrete spectrum of the three-particle Sehrgdinger operator," Mat. Sb., 9 ( l a 6 ) , No. 4 (8), 567-592 (1974). 6. Yu. N. Ovchinnikov and I. M. Sigal, "Number of bound states of three-particle systems and Efimov's effect," Ann. Physics, 123, 274-295 (1989). 7. H. Tamura, "The Efimov effect of three-body SchrSdinger operators," J. Funct. Anal., 95, 433-459 (1991). 8. D. C. Mattis, "The few-body problem on lattice," Rev. Modern Phys., 58, No. 2, 361-379 (1986). 9. V. A. Malyshev and R. A. Minlos, "Cluster operators," Trudy Sem. ira. I. G. Petrovskogo, No. 9, 63-80 (1983). 10. A. I. Mogiltner, "The problem of a few quasi-particles in solid state physics," In: Applications of Self-Adjoint Extensions in Quantum Physics (P. Exner and P. Seba eds.), Leer. Notes Phys., Vol. 324, Springer-Verlag, Berlin (1988). 11. S. N. Lakaev, "On an infinite number of three particle bound states of a system of three quantum lattice particles," Teor. Mat. Fiz., 89, No. 1, 94-104 (1991). 12. S. A. Vugalter and G. M. Zhislin, "The symmetry and Efimov's effect in systems of three quantum particles," Commun. Math. Phys., 87, 89-103 (1982). 13. M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York-San Francisco-London (1978). 14. S. N. Lakaev, "Bound states and resonances of the N-particle discrete Schr6dinger operator," Teor. Mat. Fiz., 91, No. 1 (1992). Translated by V. M. Volosov ]75
