This article is cited in 2 papers

Lower bound states of one-particle Hamiltonians on an integer lattice

Z. E. Muminov^ab, U. N. Kulzhanov<sup>a

a</sup> Samarkand State University, Samarkand, Uzbekistan
^b Uzbekistan Academy of Sciences, Samarkand Branch, Samarkand, Uzbekistan

Abstract: Under consideration is a Hamiltonian $H$ describing the motion of a quantum particle on a $d$-mentional lattice in an exterior field. It is proven that if $H$ has an eigenvalue at the lower bound of its spectrum then this eigenvalue is nondegenerate and the corresponding eigenfunction is strictly positive (thereby a lattice analog of the Perron–Frobenius theorem is proven).

Key words: spectral properties, one-particle Hamiltonian on a lattice, Birman–Schwinger principle, eigenvalue, strictly positive function.

UDC: 517.984

Received: 16.12.2010

 English version: Siberian Advances in Mathematics, 2013, 23:1, 61–68

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