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Large time behavior of the solutions to the difference wave operators

H. Islami, B. Vainberg

TL;DR

The paper addresses the Cauchy problem for the $2$-D difference wave operators on the lattice, with potentials and initial data supported in a bounded region. It determines the large-time asymptotics of solutions, highlighting differences with the analogous continuous Euclidean problem. A key finding is that the resolvent, denoted by $\mathcal{R}$, of the stationary problem has singularities on the continuous spectrum in the discrete setting, and these singularities contribute to the long-time behavior. The analysis is spectral-analytic, leveraging the structure of $\mathcal{R}$ to characterize the asymptotics and clarify the impact of the continuous spectrum on discrete lattice dynamics.

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.

Large time behavior of the solutions to the difference wave operators

TL;DR

The paper addresses the Cauchy problem for the -D difference wave operators on the lattice, with potentials and initial data supported in a bounded region. It determines the large-time asymptotics of solutions, highlighting differences with the analogous continuous Euclidean problem. A key finding is that the resolvent, denoted by , of the stationary problem has singularities on the continuous spectrum in the discrete setting, and these singularities contribute to the long-time behavior. The analysis is spectral-analytic, leveraging the structure of to characterize the asymptotics and clarify the impact of the continuous spectrum on discrete lattice dynamics.

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.

Paper Structure

This paper contains 10 theorems, 134 equations.

Key Result

Theorem 1

. For each fix \xi \in Z^{2}, Green's function G is analytic in k when k\in \overline{C}_{+}\backslash \left[ -\sqrt{8},\sqrt{8}\right] and it admits an analytic extension on \left[ -\sqrt{8},\sqrt{8}\right] \backslash \cup k_{s}. Green's function G in a neighborhood of each point k_{s} has the form where u_{1,s},u_{2,s} are analytic in k for each \xi \in Z^{2}.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Theorem 9
  • Lemma 10