Spectral properties of Schr\"{o}dinger-type operators and large-time behavior of the solutions to the corresponding wave equation

Abstract

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