On the discrete spectrum of two-particle discrete Schrödinger operators

Abstract

In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schrödinger operators $h^{\mathrm{d}}(k)=h_0^{\mathrm{d}}(k)+\mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional lattice ${\mathbb{Z}}^{\mathrm{d}},$ $\mathrm{d}\ge 1,$ $k$ being the two-particle quasi-momentum. Under some condition in the case $k\in \T^{\mathrm{d}}\setminus (-π,π)^{\mathrm{d}},$ we establish necessary and sufficient conditions for existence of infinite discrete spectrum of the operator $h^{\mathrm{d}}(k)$.

11 pages