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

## TL;DR

This paper investigates the radiation conditions for the difference Schrödinger operator on integer lattices, identifying exceptional spectral points where the limiting absorption principle fails and establishing conditions that uniquely determine solutions.

## Contribution

It characterizes the spectral points where the limiting absorption principle fails and derives radiation conditions that uniquely specify solutions for the difference Schrödinger equation on lattices.

## Key findings

- Identifies exceptional spectral points where the limiting absorption principle fails.
- Derives radiation conditions that select unique solutions outside the exceptional set.
- Describes the wave structure of solutions depending on spectral parameter.

## 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 $\lambda.$