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

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

### 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
Full Text:

### References:

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