Spectral properties of schrödinger-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 begineqnarray &(1) ddotw+ Lw=0, quad w(0)=0,quad dotw(0)=f, quad dotw=dwdt, f in H. &(2) quad ddotu+Lu=f e-ikt, quad u(0)=0, quad dotu(0)=0, endeqnarray 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 kd2 >0 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. © 2013 EDP Sciences.