The existence and location of eigenvalues of the one particle Hamiltonians on lattices
We consider a quantum particle moving in the one dimensional lattice
Z and interacting with a indefinite sign external field vˆ. We prove that
the associated Hamiltonian H can have one or two eigenvalues, situated
as below the bottom of the essential spectrum, as well as above the its
top. Moreover, we show that the operator H can have two eigenvalues
outside of the essential spectrum and one of them is situated below the
bottom of the essential spectrum, and other one above its top