Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

Abstract

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice Z 3 and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with k ∈ T 3 = (−π, π] 3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k = 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), K ∈ T 3 being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N (0, z) of eigenvalues of H(0) lying below z < 0 the following limit existsMoreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number N (K, τess(K)) of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N (K, 0) of eigenvalues lying below zero is given.

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

“…Remark 3.6. An analogue lemma has been proven in [4] in the case where the functions u(·), b(·) and w(·, ·) are analytic on T 3 and (T 3 ) 2 , respectively. )) by definition of the function D(·, ·) and Assumptions 2.1 and 2.2 we obtain that the function D(·, ζ) is of class C (2) (U δ (0)) for any ζ ∈ C + .…”

“…For instance, due to the fact that the discrete analogue of the Laplacian (or its generalizations) is not rotationally invariant, the Hamiltonian of a system does not separate into two parts, one relating to the center-of-mass motion and the other one to the internal degrees of freedom. In particular, the Efimov effect exists only for the zero value of the three-particle quasi-momentum K ∈ T 3 (see, e.g., [3,4,6,17,20,21,25] for relevant discussions and [10,11,16,25,26,28,31,38,40] for the general study of the low-lying excitation spectrum for quantum systems on lattices).…”

Existence of an isolated band in a system of three particles in an optical lattice

“…Thus, only the operator H µ0 (0) has infinitely many number of eigenvalues below the bottom of the essential spectrum (the Efimov effect) [2,6] and this result yields the existence of bound states of…”

“…Since, the operator H(K), K ∈ T 3 continuously depends on K ∈ T 3 one can conclude that there exists a neighborhood G 0 ⊂ T 3 of 0 ∈ T 3 and for all K ∈ G 0 the operator H(K), K ∈ G 0 has bound states [2,6].…”

Universality of the discrete spectrum asymptotics of the three-particle Schrödinger operator on a lattice

“…Typically, this eigenvector depends continuously on K. Therefore, Efimov's effect may exist only for some values of K ∈ T 3 . The presence of the Efimov effect for three-particle discrete Schrödinger operators was proved in [16][17][18] and asymptotic formulas for the number of eigenvalues were obtained in [16,17], which are analogous to the results of [7,9].…”