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The spectrum of the damped wave operator for a bounded domain in \(\mathbb R^2\). (English) Zbl 1061.35064

The authors consider the damped wave equation with a nonnegative potential on a Riemannian manifold and prove a necessary condition for the real parts of the eigenvalues to accumulate at a point. Then they consider the special case of a sphere and determine an interval which contains the supports of the weak limits of the sequences of probability measures which count the number of eigenvalues in a horizontal strip whose axis is the real axis, as the thickness of the strip tends to infinity. A previous result in this direction is due to J. Sj√∂strand. Then some numerical simulations follow.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J45 Hyperbolic equations on manifolds
35B37 PDE in connection with control problems (MSC2000)
93C20 Control/observation systems governed by partial differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
49J20 Existence theories for optimal control problems involving partial differential equations
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References:

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