Radiation conditions for the difference schrödinger operators

Abstract

The problem of determining a unique solution of the Schrödinger equation on the lattice is considered, where Δ is the difference Laplacian and both f and q have finite supports. It is shown that there is an exceptional set S_o of points on for which the limiting absorption priciple fails, even for unperturbed operator (q(x)=0). This exceptional set consists of the points when d is even and when d is odd. For all Values of , 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 conbinations of several waves propagating with different frequencies, and the number of waves depends on the value of λ.

Keywords:

• Schrödinger operator
• Lattice
• Limiting absorption principle
• Radiation conditions

1991 AMS Classification:

• 35P25
• 47B39

^∗Corresponding author.

^∗Corresponding author.

Notes

^∗Corresponding author.