The infiniteness of the number of eigenvalues in the gap in the essential spectrum for the three-particle Schrödinger operator on a lattice

Abstract

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via attractive pair contact potentials. We find a condition for a gap to appear in the essential spectrum and prove that there are infinitely many eigenvalues of the Hamiltonian of the corresponding three-particle system in this gap.

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

“…The main results in this section are proved the same way as in [6] in the case where the functions (potentials) v 1 ( · ), v 2 ( · ), and v 3 ( · ) are constants.…”

“…We note that the operator W 3 (z) is defined for every fixed z / ∈ σ(H 3 (K)); in particular, it is defined for every fixed z in an arbitrary gap (a, b) of the spectrum σ(H 3 (K)). We note that at the expense of the operator H 3 (K), the essential spectrum of H(K) can also contain a gap [6]. In this case, T( · ) regarded as an operator function is also well defined on the gap of the essential spectrum of H(K).…”

“…Let L 2 ((T 3 ) 2 ) be the Hilbert space of square-integrable functions defined on (T 3 ) 2 . We consider the three-particle lattice operator H(K), K ∈ T 3 , that appears in the decomposition into a direct integral for the momentum representation of the Hamiltonian of a system of three arbitrary quantum particles on the lattice Z 3 that interact via attractive short-range pair potentials [5], [6]. The operator H(K), K ∈ T 3 , acts in the Hilbert space L 2 ((T 3 ) 2 ) by the formula…”

“…This operator plays an important role in deriving the abovementioned asymptotic representation. We note that in contrast to the essential spectrum of the three-particle continuous Schrödinger operator, the essential spectrum σ ess (H(K)) of the three-particle discrete Schrödinger operator H(K), where K is the total quasimomentum of the three-particle system, may contain gaps, i.e., there can exist an interval (a, b), inf σ ess (H(K)) ≤ a < b ≤ sup σ ess (H(K)), such that (a, b) ∩ σ ess (H(K)) = ∅ (see [6]). …”

Finiteness of discrete spectrum of the two-particle Schrӧdinger operator on diamond lattices

“…The following lemma is a modification of the well-known Birman-Schwinger principle for the operator H (see [17,18] 3) and…”