Spectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation

Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA

^∗ Corresponding author. E-mail: ramm@math.ksu.edu

Abstract

Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations

$\begin{array}{ccc}& \mathrm{\left(}\mathrm{1}\mathrm{\right)} \begin{array}{}\mathrm{¨}\\ \mathit{w}\end{array}\mathrm{+}\mathit{Lw}\mathrm{=}\mathrm{0}\mathit{,} \mathit{w}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,} \mathit{ẇ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{f,} \mathit{ẇ}\mathrm{=}\frac{\mathit{dw}}{\mathit{dt}}\mathit{,} \mathit{f}\mathrm{\in }\mathit{H}\mathit{.}& \\ & \mathrm{\left(}\mathrm{2}\mathrm{\right)} \begin{array}{}\mathrm{¨}\\ \mathit{u}\end{array}\mathrm{+}\mathit{Lu}\mathrm{=}\mathit{f}{\mathit{e}}^{\mathrm{-}\mathit{ikt}}\mathit{,} \mathit{u}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,} \mathit{u̇}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}& \end{array}$

where k > 0 is a constant. Necessary and sufficient conditions are given for the operator L not to have eigenvalues in the half-plane Rez < 0 and not to have a positive eigenvalue at a given point k_d² > 0. These conditions are given in terms of the large-time behavior of the solutions to problem (1) for generic f.

Sufficient conditions are given for the validity of a version of the limiting amplitude principle for the operator L.

A relation between the limiting amplitude principle and the limiting absorption principle is established.