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Trace formula for resonances in small domains. (English) Zbl 1068.47055

Summary: Trace formulas are useful in the study of resonances. In this article, we work in a semi-classical black-box framework and obtain a trace formula of the same type as that obtained by the second author previously, now for the resonances in \(h\)-dependent domains of size \(h\), and under somewhat more restrictive assumptions. As an application, we extend to a long-range situation a result by V. M. Petkov and M. Zworski [Commun. Math. Phys. 204, No. 2, 329–351 (1999; Zbl 0936.47004); erratum ibid. 214, No. 3, 733–735 (2000; Zbl 1046.81551)] on Breit-Wigner peaks in the scattering phase.

MSC:

47F05 General theory of partial differential operators
35P25 Scattering theory for PDEs
30E10 Approximation in the complex plane
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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