On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 3, page 625-635
  • ISSN: 0373-0956

Abstract

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We consider selfadjoint positively definite operators of the form - Δ + P (not necessarily elliptic) in ℝ n , n ≥ 3 , odd, where P is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if { λ j } ( Im λ j ≥ 0 ) are the scattering poles associated to the operator - Δ + P repeated according to multiplicity, it is proved that for any ϵ > 0 there exists a constant C ϵ > 0 so that # { λ j : | λ j | ≤ r , ϵ ≤ arg λ j ≤ π - ϵ } ≤ C ϵ r n for any r ≥ 1 .

How to cite

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Vodev, Georgi. "On the distribution of scattering poles for perturbations of the Laplacian." Annales de l'institut Fourier 42.3 (1992): 625-635. <http://eudml.org/doc/74967>.

@article{Vodev1992,
abstract = {We consider selfadjoint positively definite operators of the form $- \Delta +P$ (not necessarily elliptic) in $\{\Bbb R\}^ n$, $n\ge 3$, odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if $\lbrace \lambda _ j\rbrace (\operatorname\{Im \}\lambda _ j\ge 0_ )$ are the scattering poles associated to the operator $- \Delta +P$ repeated according to multiplicity, it is proved that for any $\varepsilon &gt;0$ there exists a constant $C_ \varepsilon &gt;0$ so that $\#\lbrace \lambda _ j:\vert \lambda _ j\vert \le r$, $\varepsilon \le \arg \lambda _ j\le \pi -\varepsilon \rbrace \le C_ \varepsilon r^ n$ for any $r\ge 1$.},
author = {Vodev, Georgi},
journal = {Annales de l'institut Fourier},
keywords = {scattering poles; cutoff resolvent},
language = {eng},
number = {3},
pages = {625-635},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the distribution of scattering poles for perturbations of the Laplacian},
url = {http://eudml.org/doc/74967},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Vodev, Georgi
TI - On the distribution of scattering poles for perturbations of the Laplacian
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 3
SP - 625
EP - 635
AB - We consider selfadjoint positively definite operators of the form $- \Delta +P$ (not necessarily elliptic) in ${\Bbb R}^ n$, $n\ge 3$, odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if $\lbrace \lambda _ j\rbrace (\operatorname{Im }\lambda _ j\ge 0_ )$ are the scattering poles associated to the operator $- \Delta +P$ repeated according to multiplicity, it is proved that for any $\varepsilon &gt;0$ there exists a constant $C_ \varepsilon &gt;0$ so that $\#\lbrace \lambda _ j:\vert \lambda _ j\vert \le r$, $\varepsilon \le \arg \lambda _ j\le \pi -\varepsilon \rbrace \le C_ \varepsilon r^ n$ for any $r\ge 1$.
LA - eng
KW - scattering poles; cutoff resolvent
UR - http://eudml.org/doc/74967
ER -

References

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  14. [14] G. VODEV, Sharp bounds on the number of scattering poles for perturbations of the Lapacian, Commun. Math. Phys., 145 (1992), to appear. Zbl0766.35032MR1191913
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