Radiation Conditions for the Difference Schr\"{o}dinger Operators

Abstract

The problem of determining a unique solution of the Schr\"{o}dinger equation $\left(\Delta+q-\lambda\right) \psi=f$ on the lattice $\mathbb{Z}^{d}$ is considered, where $\Delta$ is the difference Laplacian and both $f$ and $q$ have finite supports$.$ It is shown that there is an exceptional set $S_{0}$ of points on $Sp(\Delta)=[-2d,2d]$ for which the limiting absorption principle fails, even for unperturbed operator ($q(x)=0$). This exceptional set consists of the points $\left\{ \pm4n\right\} $ when $d$ is even and $\left\{ \pm2(2n+1)\right\} $ when $d$ is odd. For all values of $\lambda \in[-2d,2d]\backslash S_{0},$ the radiation conditions are found which single out the same solutions of the problem as the ones determined by the limiting absorption principle. These solutions are combinations of several waves propagating with different frequencies, and the number of waves depends on the value of $\lambda.$