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Sharp Upper Bounds on the Number of Resonances near the Real Axis for Trapping Systems

Stefanov, Plamen

American journal of mathematics, 2003-02, Vol.125 (1), p.183-224 [Periódico revisado por pares]

Baltimore, MD: Johns Hopkins University Press

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Título:
Sharp Upper Bounds on the Number of Resonances near the Real Axis for Trapping Systems
Autor: Stefanov, Plamen
Assuntos: Classical and quantum physics: mechanics and fields ; Classical mechanics of continuous media: general mathematical aspects ; Differential operators ; Eigenfunctions ; Eigenvalues ; Energy levels ; Exact sciences and technology ; Mathematical analysis ; Mathematical models ; Mathematics ; Nonlinear dynamics and nonlinear dynamical systems ; Operator theory ; Physics ; Polynomials ; Real lines ; Resonance scattering ; Sciences and techniques of general use ; Spectral theory ; Statistical physics, thermodynamics, and nonlinear dynamical systems ; Theory ; Wave fronts ; Waves and wave propagation: general mathematical aspects
É parte de: American journal of mathematics, 2003-02, Vol.125 (1), p.183-224
Descrição: We study resonances near the real axis$(|{\rm Im}\,z|=O(h^{N}),N\gg \ 1)$and the corresponding resonant states for semiclassical long range operators P(h). Without a priori assumptions on the distribution or on the multiplicities of the resonances. we show that the truncated resonant states form a family of quasimode states for P(h), stable under small perturbations. As a consequence, they also form a family of quasimode states for any suitably defined (self-adjoint) reference operator$P^{\#}(h)$, therefore, those resonances are perturbed eigenvalues of$P^{\#}(h)$. Next we show that the semiclassical wave front set of the resonant states is contained in the set of trapped directions${\cal T}$. We construct a suitable reference operator from P(h) by imposing a microlocal barrier outside${\cal T}$to show that the counting function for those resonances admits an upper bound of Weyl's type connected with the measure of${\cal T}$. We give an example of a system for which this bound is optimal and also prove similar bounds in case of classical scattering by obstacle.
Editor: Baltimore, MD: Johns Hopkins University Press
Idioma: Inglês

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Sharp Upper Bounds on the Number of Resonances near the Real Axis for Trapping Systems

Stefanov, Plamen

Baltimore, MD: Johns Hopkins University Press

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