We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator . We assume the existence of another self-adjoint operator for which the family extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for in the vicinity of the eigenvalue, we prove that the essential spectrum is locally deformed away from the eigenvalue, leaving it isolated and thus permitting an application of Kato’s analytic perturbation theory.
Nous construisons dans cet article une théorie de perturbation analytique pour des valeurs propres avec multiplicités finies, plongées dans le spectre essentiel d’un opérateur auto-adjoint . Pour pouvoir faire ça on suppose l’existence d’un autre opérateur auto-adjoint pour lequel la famille a une extension analytique de la ligne réelle à une bande dans le plan complexe. En supposant que l’estimation de Mourre soit vraie pour au voisinage de la valeur propre, on montre que le spectre essentiel est localement déformé afin qu’il ne contienne plus la valeur propre permettant ainsi l’application de la théorie de la perturbation analytique de Kato.
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Keywords: Analytic perturbation theory, spectral deformation, Mourre theory
Mot clés : Théorie de la perturbation analytique, Déformation spectrale, Théorie de Mourre
@article{AIF_2018__68_2_767_0, author = {Engelmann, Matthias and M{\o}ller, Jacob Schach and Rasmussen, Morten Grud}, title = {Local {Spectral} {Deformation}}, journal = {Annales de l'Institut Fourier}, pages = {767--804}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {2}, year = {2018}, doi = {10.5802/aif.3177}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3177/} }
TY - JOUR AU - Engelmann, Matthias AU - Møller, Jacob Schach AU - Rasmussen, Morten Grud TI - Local Spectral Deformation JO - Annales de l'Institut Fourier PY - 2018 SP - 767 EP - 804 VL - 68 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3177/ DO - 10.5802/aif.3177 LA - en ID - AIF_2018__68_2_767_0 ER -
%0 Journal Article %A Engelmann, Matthias %A Møller, Jacob Schach %A Rasmussen, Morten Grud %T Local Spectral Deformation %J Annales de l'Institut Fourier %D 2018 %P 767-804 %V 68 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3177/ %R 10.5802/aif.3177 %G en %F AIF_2018__68_2_767_0
Engelmann, Matthias; Møller, Jacob Schach; Rasmussen, Morten Grud. Local Spectral Deformation. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 767-804. doi : 10.5802/aif.3177. http://www.numdam.org/articles/10.5802/aif.3177/
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