This article is cited in 15 papers

Spectrum of the two-particle Schrödinger operator on a lattice

S. N. Lakaev^a, A. M. Khalkhuzhaev<sup>b

a</sup> A. Navoi Samarkand State University
^b Institute of Study of Regional Problems, Uzbekistan Academy of Sciences, Samarkand Branch

Abstract: We consider the family of two-particle discrete Schrödinger operators $H(k)$ associated with the Hamiltonian of a system of two fermions on a $\nu$-dimensional lattice $\mathbb Z^{\nu}$, $\nu\geq 1$, where $k\in\mathbb T^{\nu}\equiv(-\pi,\pi]^{\nu}$ is a two-particle quasimomentum. We prove that the operator $H(k)$, $k\in\mathbb T^{\nu}$, $k\ne0$, has an eigenvalue to the left of the essential spectrum for any dimension $\nu=1,2,\dots$ if the operator $H(0)$ has a virtual level ($\nu=1,2$) or an eigenvalue ($\nu\geq 3$) at the bottom of the essential spectrum (of the two-particle continuum).

Keywords: spectral properties, two-particle discrete Schrödinger operator, Birman–Schwinger principle, virtual level, eigenvalue.

Received: 20.12.2005
Revised: 24.07.2007

DOI: 10.4213/tmf6211

 English version: Theoretical and Mathematical Physics, 2008, 155:2, 754–765

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