Counting scattering poles. (English) Zbl 0823.35139

Zworski, M.

Ikawa, Mitsuru (ed.), Spectral and scattering theory. Proceedings of the Taniguchi international workshop, held at Sanda, Hyogo, Japan. Basel: Marcel Dekker. Lect. Notes Pure Appl. Math. 161, 301-331 (1994).

This survey describes the recent results on estimating the number of scattering poles, obtained by the author and J. Sjöstrand in joint works. For compactly supported perturbations of the Laplacian in \(\mathbb{R}^ n\), various upper bounds are obtained for the number of scattering poles inside conic-type regions of the complex plane, asymptotically as the diameter of these regions tends to infinity. Many other questions are discussed, such as the distribution of the poles near the real axis, and the obtention of lower bounds on their number. Also, several examples are discussed showing the accuracy (and sometimes the optimality) of the general results.
For the entire collection see [Zbl 0798.00016].

Reviewer: A.Martinez (Villetaneuse)

MSC:

35P25	Scattering theory for PDEs
81U20	\(S\)-matrix theory, etc. in quantum theory
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations

Keywords:

number of scattering poles; upper bounds; conic-type regions; distribution of the poles; lower bounds