Spectral properties of Schrödinger-type operators and large-time behavior of the solutions to the corresponding wave equation

2012/06/26 by A. G. Ramm, Ramm, A. G.
Mathematics · #35L90 #35P25 #43A32 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1206.5990

openalex publication_date 2012/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) w+ Lw=0, w(0)=0, w(0)=f, w=(dw)/(dt), f ∈ H. &(2) u+Lu=f e^-ikt, u(0)=0, 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 Rez<0 and not to have a positive eigenvalue at a given point k_d² >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.

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