Spectral analysis of a class of Schrodinger difference operators

Sufficient discreteness conditions for the spectrum of the operator is given. Necessary and sufficient conditions for the unique solvability of the inverse problem of reconstructing the difference equation from two spectra are also obtained. Using the relationship between the Schrödinger and Dirac difference operators it is proved that the spectrum of the operator is purely discrete. A Jacobian matrix is considered and the first truncated matrix obtained by deleting the first row and the first column of the matrix is introduced. For definiteness, it is assumed that the spectra completely determine the spectral function of the operator. Two sequences of real numbers are the two spectra of some equation if and only if the numbers satisfy a certain inequality.