The damped wave equation is closely related to non-self-adjoint perturbations of a self-adjoint operator P of the form
$$\displaystyle P_\epsilon =P+i\epsilon Q. $$
Here, P is a semi-classical pseudodifferential operator of order 0 on L2(X), where we consider two cases:
X = Rn and P has the symbol P ∼ p(x, ξ) + hp1(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.
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,
Equation des ondes amorties. Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), pp. 73–109. Mathematical Physics Studies, vol. 19 (Kluwer Academic Publishers, Dordrecht, 1996)
Introduction to the Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H.H. McFaden. Translation ed. by B. Silver. With an appendix by M.V. Keldysh. Translations of Mathematical Monographs, vol. 71 (American Mathematical Society, Providence, 1988)
A.S. Markus, V.I. Matseev, Asymptotic behavior of the spectrum of close-to-normal operators. Funktsional. Anal. i Prilozhen.
13(3), 93–94 (1979), Functional Anal. Appl.
13(3), 233–234 (1979) (1980)
G. Rivière, Delocalization of slowly damped eigenmodes on Anosov manifolds. Commun. Math. Phys.
316(2), 555–593 (2012)
J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations. Publ. RIMS Kyoto Univ.
36(5), 573–611 (2000)