Note on the spectrum of discrete Schr\"odinger operators

The spectrum of discrete Schr"odinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the absence of singular continuous spectrum is proven. In particular it is shown that an embedded eigenvalue does appear for $d\ge 5$ but does not for $1\le d\le 4$.