  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 $d\geq5$ but does not for
$1\leq d\leq 4$.
