Eigenmodes of the damped wave equation and small hyperbolic subsets. (Modes propres de l’équation des ondes amorties et petits sous-ensembles hyperboliques.) (English. French summary) Zbl 1314.58019
Let \((M,g)\) be a smooth compact, connected Riemanian manifold without boundary of dimension \(d \geq 2\). In this paper the author considers the solutions \(u_\tau \in L^2(M)\) of the equation
\[
(-\Delta_g -\tau^2 -2_{ia(x)\tau})u_\tau =0, a\geq 0
\]
(which come from the solutions of the damped wave equations of the form \(v = e^{-it \tau} u_\tau\)). Motivated by earlier results by J. A. Toth and S. Zelditch [Ann. Henri Poincaré 4, No. 2, 343–368 (2003; Zbl 1028.58028)], N. Burq and M. Zworski [J. Am. Math. Soc. 17, No. 2, 443–471 (2004; Zbl 1050.35058)], H. Christianson [J. Funct. Anal. 246, No. 2, 145–195 (2007; Zbl 1119.58018)], the author studies the concentration in shrinking tubes of size \(hv, h \, 1/\tau\) (where \(0 < v < 1/2\)) around more general hyperbolic subsets than in the above cited works, under conditions of negative topological pressure. The method of proofs uses ideas from the works by N. Anantharaman and S. Nonnenmacher [Ann. Inst. Fourier 57, No. 7, 2465–2523 (2007; Zbl 1145.81033)], S. Nonnenmacher and M. Zworski [Acta Math. 203, No. 2, 149–233 (2009; Zbl 1226.35061)] and N. Anantharaman [Geom. Funct. Anal. 20, No. 3, 593–626 (2010; Zbl 1205.35173)].
This article includes also an appendix by S. Nonnenmacher and the author where they establish the existence of an inverse logarithmic strip without eigenvalues below the real axis under a prensure condition on the set of undamped trajectories.
This article includes also an appendix by S. Nonnenmacher and the author where they establish the existence of an inverse logarithmic strip without eigenvalues below the real axis under a prensure condition on the set of undamped trajectories.
Reviewer: Claude Zuily (Orsay)
MSC:
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
Citations:
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