The Threshold effects for the two-particle Hamiltonians on lattices

Abstract

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice ${\mathbb{Z}}^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

References in corpus (1)

• Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics

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