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

DOI10.1051/MMNP/20138116arXiv1206.5990MaRDI QIDQ6233959

A. G. Ramm

Publication date: 26 June 2012

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.

Mathematics Subject Classification ID

Scattering theory for PDEs (35P25) Other transforms and operators of Fourier type (43A32) Abstract hyperbolic equations (35L90)

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