Weyl Asymptotics for the Damped Wave Equation

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 L ²(X), where we consider two cases:

• 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.

• 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

  $$\displaystyle p(x,\xi )=\sum _{|\alpha |\le m}a_{\alpha ,0} (x)\xi ^\alpha . $$