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Spectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation
Published online by Cambridge University Press: 28 January 2013
Abstract
Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations
\begin{eqnarray} &(1) \quad \ddot{w}+ Lw=0, \quad
w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. \\ &(2)
\quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0,
\end{eqnarray}
(1) ¨w+Lw=0, w(0)=0, ẇ(0)=f, ẇ=dwdt, f∈H.(2) ¨u+Lu=fe−ikt, u(0)=0, u̇(0)=0,
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 kd2 > 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.
- Type
- Research Article
- Information
- Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 207 - 214
- Copyright
- © EDP Sciences, 2013
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