Dispersive estimates and absence of embedded eigenvalues

Herbert Koch; Daniel Tataru

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This text contains a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations. Part III, in collaboration with B. Helffer, gives explicit estimates in the Gearhardt-Prüss theorem for semi-groups.

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We study semiclassical resonances in a box $Ω (h)$ of height $h^{N}$ , $N ≫ 1$ . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $𝒯$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^{#} (h)$ with discrete spectrum the number of resonances in $Ω (h)$ is bounded by the number of eigenvalues of $P^{#} (h)$ in an interval a bit larger than the projection of $Ω (h)$ on the real line. As an application,...