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

TL;DR

This paper investigates the spectral properties of Schrödinger-type operators and their influence on the long-term behavior of solutions to related wave equations, establishing conditions for eigenvalue absence and principles like limiting amplitude.

Contribution

It provides necessary and sufficient conditions for eigenvalue absence in certain half-planes and links the limiting amplitude principle with the limiting absorption principle for these operators.

Findings

Conditions for absence of eigenvalues in Re z<0

Criteria for positive eigenvalues at specific points

Relations between limiting amplitude and absorption principles

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.