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Published in:
2019 | OriginalPaper | Chapter
14. Weyl Asymptotics for the Damped Wave Equation
Author : Johannes Sjöstrand
Published in: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
Publisher: Springer International Publishing
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Abstract
The damped wave equation is closely related to non-self-adjoint perturbations of a self-adjoint operator P of the form Here, P is a semi-classical pseudodifferential operator of order 0 on L
2(X), where we consider two cases:
$$\displaystyle P_\epsilon =P+i\epsilon Q. $$
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X = R n and P has the symbol P ∼ p(x, ξ) + hp 1(x, ξ) + ⋯ . in S(m), as in Sect. 6.1, where the description is valid also in the case n > 1. We assume for simplicity that the order function m(x, ξ) tends to + ∞, when (x, ξ) tends to ∞. We also assume that P is formally self-adjoint. Then by elliptic theory (and the ellipticity assumption on P) we know that P is essentially self-adjoint with purely discrete spectrum.
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X is a compact smooth manifold with positive smooth volume form dx and P is a formally self-adjoint differential operator, which in local coordinates takes the form,where \(a_\alpha (x;h)\sim \sum _{k=0}^\infty h^ka_{\alpha ,k}(x)\) in C ∞ and the “classical” principal symbol$$\displaystyle P=\sum _{|\alpha |\le m}a_\alpha (x;h)(hD_x)^\alpha ,\ m>0 $$satisfies$$\displaystyle p_m(x,\xi )=\sum _{|\alpha |=m}a_{\alpha ,0} (x)\xi ^\alpha , $$so m has to be even. In this case the semi-classical principal symbol is given by$$\displaystyle 0\le p_m(x,\xi )\asymp |\xi |{ }^m , $$$$\displaystyle p(x,\xi )=\sum _{|\alpha |\le m}a_{\alpha ,0} (x)\xi ^\alpha . $$