Geodesic
Spectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation
A. G. Ramm1
1 Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 1, pp. 207-214

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Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations \begin{eqnarray} (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, \end{eqnarray} ( 1 )   ¨ w + Lw = 0 ,   w ( 0 ) = 0 ,   ẇ ( 0 ) = f,   ẇ = dw dt ,   f ∈ H . ( 2 )   ¨ u + Lu = f e − ikt ,   u ( 0 ) = 0 ,   u̇ ( 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 kd2 > 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.
DOI : 10.1051/mmnp/20138116

A. G. Ramm 1

1 Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA 
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A. G. Ramm. Spectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 1, pp. 207-214. doi: 10.1051/mmnp/20138116

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