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A trace formula and review of some estimates for resonances. (English) Zbl 0877.35090
Rodino, Luigi (ed.), Microlocal analysis and spectral theory. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Castelvecchio Pascoli (Lucca), Italy, 23 September–3 October 1996. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 490, 377-437 (1997).
In these lecture notes, the author establishes and gives applications of a new trace formula involving a pair of operators: This formula links the trace of their difference with their possible resonances, and actually applies also when the operators do not act on the same Hilbert space. As a first application, the author obtains a lower bound on the number of scattering poles, which applies e.g. to the Laplacian with obstacle in $$\mathbb{R}^n$$ ($$n\geq 2$$ arbitrary); this result generalizes a previous one obtained in a joint work with M. Zworski. Then the author reviews some results on the semiclassical Schrödinger operator (obtained with Zworksi), and concerning upper bounds on the density of resonances near the real axis.
For the entire collection see [Zbl 0864.00062].

##### MSC:
 35P25 Scattering theory for PDEs
##### Keywords:
scattering poles; density of resonances