*Math. Model. Nat. Phenom.*Vol. 8, No. 1, 2013, pp. 207-214

## 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

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
Re*z* < 0 and not to have a positive eigenvalue at a given point *k*_{d}^{2} > 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

*© EDP Sciences, 2013*