The Existence and location of eigenvalues of the one particle Hamiltonians on lattices

We consider a quantum particle moving in the one dimensional latticeZ and interacting with a indefinite sign external field vˆ. We prove thatthe associated Hamiltonian H can have one or two eigenvalues, situatedas below the bottom of the essential spectrum, as well as above the itstop. Moreover, we show that the operator H can have two eigenvaluesoutside of the essential spectrum and one of them is situated below thebottom of the essential spectrum, and other one above its top

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