Geodesic
Lower bound states of one-particle Hamiltonians on an integer lattice
Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 129-140.

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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). 
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Z. E. Muminov; U. N. Kulzhanov. Lower bound states of one-particle Hamiltonians on an integer lattice. Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 129-140. http://geodesic.mathdoc.fr/item/MT_2012_15_1_a7/

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