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Spectral properties of a two-particle Hamiltonian on a lattice. (English. Russian original) Zbl 1298.81087

Theor. Math. Phys. 177, No. 3, 1693-1705 (2013); translation from Teor. Mat. Fiz. 177, No. 3, 468-481 (2013).
Summary: We consider a system of two arbitrary quantum particles on a three-dimensional lattice with some dispersion functions (describing particle transport from a site to a neighboring site). The particles interact via an attractive potential at only the nearest-neighbor sites. We study how the number of eigenvalues of a family of operators \(h(k)\) depends on the particle interaction energy and the total quasimomentum \(k\in\mathbb T^3\), where \(\mathbb T^3\) is a three-dimensional torus. We find the conditions under which the operator \(h(0)\) has a double or triple virtual level at zero depending on the particle interaction energy.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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