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. Sjöstrand. Then some numerical simulations follow.


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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