Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Plamen Stefanov

Journées équations aux dérivées partielles (2001)

  • page 1-16
  • ISSN: 0752-0360

Abstract

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We study semiclassical resonances in a box Ω ( h ) of height h N , N ≫ 1 . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set 𝒯 of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # ( h ) with discrete spectrum the number of resonances in Ω ( h ) is bounded by the number of eigenvalues of P # ( h ) in an interval a bit larger than the projection of Ω ( h ) on the real line. As an application, we prove a Weyl type estimate of the number of resonances in Ω ( h ) in terms of the measure of 𝒯 . We prove a similar estimate in case of classical scattering by a metric and obstacle.

How to cite

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Stefanov, Plamen. "Weyl type upper bounds on the number of resonances near the real axis for trapped systems." Journées équations aux dérivées partielles (2001): 1-16. <http://eudml.org/doc/93410>.

@article{Stefanov2001,
abstract = {We study semiclassical resonances in a box $\Omega (h)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $\mathcal \{T\}$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^\#(h)$ with discrete spectrum the number of resonances in $\Omega (h)$ is bounded by the number of eigenvalues of $P^\#(h)$ in an interval a bit larger than the projection of $\Omega (h)$ on the real line. As an application, we prove a Weyl type estimate of the number of resonances in $\Omega (h)$ in terms of the measure of $\mathcal \{T\}$. We prove a similar estimate in case of classical scattering by a metric and obstacle.},
author = {Stefanov, Plamen},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-16},
publisher = {Université de Nantes},
title = {Weyl type upper bounds on the number of resonances near the real axis for trapped systems},
url = {http://eudml.org/doc/93410},
year = {2001},
}

TY - JOUR
AU - Stefanov, Plamen
TI - Weyl type upper bounds on the number of resonances near the real axis for trapped systems
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 16
AB - We study semiclassical resonances in a box $\Omega (h)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $\mathcal {T}$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^\#(h)$ with discrete spectrum the number of resonances in $\Omega (h)$ is bounded by the number of eigenvalues of $P^\#(h)$ in an interval a bit larger than the projection of $\Omega (h)$ on the real line. As an application, we prove a Weyl type estimate of the number of resonances in $\Omega (h)$ in terms of the measure of $\mathcal {T}$. We prove a similar estimate in case of classical scattering by a metric and obstacle.
LA - eng
UR - http://eudml.org/doc/93410
ER -

References

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