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We construct multiple families of solitary standing waves of the discretecubically nonlinear Schr\"{o}dinger equation (DNLS) in dimensions $d=1,2$ and$3$. These states are obtained via a bifurcation analysis about the continuum(NLS) limit. One family consists {\it on-site symmetric} (vertex-centered)states; these are spatially localized solitary standing waves which aresymmetric about any fixed lattice site. The other spatially localized statesare {\it off-site symmetric}. Depending on the spatial dimension, these may bebond-centered, cell-centered, or face-centered. Finally, we show that theenergy difference among distinct states of the same frequency is exponentiallysmall with respect to a natural parameter. This provides a rigorous bound forthe so-called {\it Peierls-Nabarro} energy barrier.

On-Site and Off-Site Bound States of the Discrete Nonlinear Schr\"odinger Equation and the Peierls-Nabarro Barrier