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
∗ Corresponding author. E-mail: email@example.com
Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations
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.
Mathematics Subject Classification: 35P25 / 35L90 / 43A32
Key words: elliptic operators / wave equation / limiting amplitude principle / limiting absorption principle
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