Spectral instability of some non-selfadjoint anharmonic oscillators

Raphaël Henry

doi:10.1016/j.crma.2012.11.011

Partial Differential Equations

Spectral instability of some non-selfadjoint anharmonic oscillators

Presented by: the Editorial Board

Raphaël Henry ¹

¹ Université Paris-Sud, 91405 Orsay cedex, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1043-1046.

Abstract
Résumé

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $Π_{n}$ associated with their eigenvalues. More precisely, we are concerned with some anharmonic oscillators $A (m, θ) = - \frac{d^{2}}{d x^{2}} + e^{i θ} {| x |}^{m}$ with $| θ | < \min {\frac{(m + 2) π}{4}, \frac{(m + 2) π}{2 m}}$ , defined on $L^{2} (R)$ . We get asymptotic expansions for the norm of the spectral projections associated with the large eigenvalues of $A (1, θ)$ and $A (2 k, θ)$ , $k ⩾ 1$ , extending the results of Davies (2000) [4] and Davies and Kuijlaars (2004) [5].

Notre objectif est de mettre en évidence lʼinstabilité spectrale de certains opérateurs différentiels non-autoadjoints, via lʼétude de la croissance des normes des projecteurs spectraux $Π_{n}$ associés à leurs valeurs propres. Nous nous intéressons à certains oscillateurs anharmoniques $A (m, θ) = - \frac{d^{2}}{d x^{2}} + e^{i θ} {| x |}^{m}$ avec $| θ | < \min {\frac{(m + 2) π}{4}, \frac{(m + 2) π}{2 m}}$ , définis sur $L^{2} (R)$ . Nous étendons les résultats de Davies (2000) [4] et Davies et Kuijlaars (2004) [5] en donnant un développement asymptotique de la norme des projecteurs spectraux associés aux grandes valeurs propres pour les opérateurs $A (1, θ)$ et $A (2 k, θ)$ , $k ⩾ 1$ .

Received: 2012-11-05
Accepted: 2012-11-19
Published online: 2012-11-27

DOI: 10.1016/j.crma.2012.11.011

Author's affiliations:

Raphaël Henry ¹

¹ Université Paris-Sud, 91405 Orsay cedex, France

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     author = {Rapha\"el Henry},
     title = {Spectral instability of some non-selfadjoint anharmonic oscillators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1043--1046},
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     year = {2012},
     doi = {10.1016/j.crma.2012.11.011},
     language = {en},
}

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Raphaël Henry. Spectral instability of some non-selfadjoint anharmonic oscillators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 23-24, pp. 1043-1046. doi : 10.1016/j.crma.2012.11.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.11.011/

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