On the discrete spectrum of two-particle discrete Schrödinger operators

Published 16 Mar 2011 in math.FA | (1103.3132v1)

Abstract: In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{{\mathrm{d}}(k)=h^{{\mathrm{d}}_0(k)+}} \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional lattice $\Z^{{\mathrm{d}},$} $\mathrm{d}\geq 1,$ $k$ being the two-particle quasi-momentum. Under some condition in the case $k\in \T^{{\mathrm{d}}\setminus} (-\pi,\pi)^{{\mathrm{d}},$} we establish necessary and sufficient conditions for existence of infinite discrete spectrum of the operator $h^{{\mathrm{d}}(k)$.}