# Weyl Asymptotics for the Damped Wave Equation

• Johannes Sjöstrand
Chapter
Part of the Pseudo-Differential Operators book series (PDO, volume 14)

## Abstract

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. , 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,
$$\displaystyle P=\sum _{|\alpha |\le m}a_\alpha (x;h)(hD_x)^\alpha ,\ m>0$$
where $$a_\alpha (x;h)\sim \sum _{k=0}^\infty h^ka_{\alpha ,k}(x)$$ in C and the “classical” principal symbol
$$\displaystyle p_m(x,\xi )=\sum _{|\alpha |=m}a_{\alpha ,0} (x)\xi ^\alpha ,$$
satisfies
$$\displaystyle 0\le p_m(x,\xi )\asymp |\xi |{ }^m ,$$
so m has to be even. In this case the semi-classical principal symbol is given by
$$\displaystyle p(x,\xi )=\sum _{|\alpha |\le m}a_{\alpha ,0} (x)\xi ^\alpha .$$

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