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Sjöstrand, Johannes

Geometric bounds on the density of resonances for semiclassical problems. (English) Zbl 0702.35188

Duke Math. J. 60, No. 1, 1-57 (1990).
The author gives upper bounds on the number of resonances in certain regions in the complex plane close to the real axis, for semi-classical operators \(like:-h^ 2\Delta +V(x)\) when h is small. A part of the results was announced in the Proc. of the VIII Latin American School of Mathematics (1986) [Lect. Notes Math. 1324, 286-292 (1988; Zbl 0674.35018)]. The resonances are defined using a microlocal version of the complex scaling due to the reviewer and the author [Mém. Soc. Math. Fr., Nouv. Sér. 24/25 (1986; Zbl 0631.35075)] and the results presented here are analogous to those by R. Melrose [Journ. “Equations Deriv. Partielles”, St. Jean-De Monts 1984, Conf. No.3, 8 p. (1984; Zbl 0621.35073)] in the context of the exterior problem.
Reviewer: B.Helffer

Cited in 4 Reviews
Cited in 59 Documents

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation

Keywords:

resonances; semi-classical operators; complex scaling

Citations:

Zbl 0674.35018; Zbl 0631.35075; Zbl 0621.35073
Cite Review PDF
Full Text: DOI

References:

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