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 spectrumthen 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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