Lower bound states of one-particle Hamiltonians on an integer lattice. (Russian, English) Zbl 1340.81012
Mat. Tr. 15, No. 1, 129-140 (2012); translation in Sib. Adv. Math. 23, No. 1, 61-68 (2013).
Summary: Under consideration is a Hamiltonian \(H\) describing the motion of a quantum particle on a \(d\)-dimensional 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).
MSC:
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
35P20 | Asymptotic distributions of eigenvalues in context of PDEs |