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Sharp upper bounds on the number of resonances near the real axis for trapping systems. (English) Zbl 1040.35055

The main goal of the paper is to obtain sharp upper bounds on the number of resonances for second order operators \(P(h)\) which are compactly supported perturbations of the semiclassical long range Schrödinger operator. The method used by the author to get such estimates is based on the fact that the number of resonances near the real axis is less or equal with the number of eigenvalues of an appropriate reference operator. The author defines rigourously the notion of reference operators and proves that for such operators the assertion from above is true. This result holds in the general framework of black box scattering proposed by J. Sjöstrand and M. Zworski [J. Am. Math. Soc. 4, 729–769 (1991; Zbl 0752.35046)].
The definition which is given for the reference operator allows some flexibility in its choice. For the estimate of the number of resonances of \(P(h)\) near the real axis the author considers a reference operator as a perturbation of \(P(h)\) with a \(h\)-pseudodifferential operator with principal symbol that vanishes near the trapped rays and increases quickly outside some small neighborhood of them. This allows to get Weyl type upper bounds for the number of resonances. An example where the upper bound turns into an asymptotic formula is given. Weyl type estimates of the number of resonances in case of classical scattering by an obstacle are also proved.

MSC:

35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
35B34 Resonance in context of PDEs
81U05 \(2\)-body potential quantum scattering theory

Citations:

Zbl 0752.35046
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