Large Time Behavior of Solutions to Difference Wave Operators

ABSTRACT

The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case (when the problem in the Euclidian space is considered, not on the lattice) the resolvent of the corresponding stationary problem has singularities on the continuous spectrum, and they contribute to the asymptotics.

Keywords:

• Cauchy problem
• Difference wave operator
• Large time asymptotics
• Lattice

Mathematics Subject Classification:

• 39A11
• 35L10

Acknowledgment

The work by B. Vainberg was supported partially by the NSF grant DMS-0405927.