On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

On the d -dimensional lattice ℤ^d and the r -regular tree T^r , an exact expression for the number of discrete eigenvalues of a discrete Laplacian with a finitely supported potential is described in terms of the support and the intensities of the potential on each case. In particular, the number of eigenvalues less than the infimum of the essential spectrum is bounded by the number of negative intensities.