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

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

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