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

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).