Local Spectral Deformation

Engelmann, Matthias; Møller, Jacob Schach; Rasmussen, Morten Grud

Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 767-804.

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 $H$. Pour pouvoir faire ça on suppose l’existence d’un autre opérateur auto-adjoint $A$ pour lequel la famille $H_{\theta }=e^{\mathrm{i}\theta A}H{e}^{-\mathrm{i}\theta A}$ 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 $\mathrm{i}[H,A]$ 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.

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family $H_{\theta }=e^{\mathrm{i}\theta A}H{e}^{-\mathrm{i}\theta A}$ extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for $\mathrm{i}[H,A]$ 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.

Classification : 81Q10, 47A55, 81Q12
Keywords: Analytic perturbation theory, spectral deformation, Mourre theory
Mot clés : Théorie de la perturbation analytique, Déformation spectrale, Théorie de Mourre

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