The Threshold effects for the two-particle Hamiltonians on lattices

Details The Threshold effects for the two-particle Hamiltonians on lattices The Threshold effects for the two-particle Hamiltonians on lattices

2005-01-06

arxiv.org/abs/math-ph/0501013v1

Physics

Mathematical Physics

Scientific paper

10.1007/s00220-005-1454-y

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ 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.

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