Estabilidad espectral y resonancias para perturbaciones de rango finito y singulares: Spectral stability and resonances for finite rank and singular perturbations

M. Angélica Astaburuaga; Víctor H. Cortés; Claudio Fernández; Rafael Del Río

doi:10.56754/0719-0646.2702.505

DOI:

https://doi.org/10.56754/0719-0646.2702.505

Abstract

In these notes, we summarize a series of papers devoted to perturbations of operators of several classes, among them differential operators. The articles mentioned before, study spectral properties, with special emphasis on the stability of the eigenvalues and the absence of a certain singular spectrum. These perturbations are of a different nature, including finite rank and the singular case. We also characterize and prove the resonance phenomenon from a dynamical point of view, that is, the existence of states with long life and for which the survival amplitude has an almost exponential behavior. In addition, we include a discussion about several open problems in the area.

Resumen

En estas notas resumimos una serie de artículos dedicados a perturbaciones de operadores de variadas clases, entre ellos operadores diferenciales. En dichos artículos se estudian propiedades espectrales, con énfasis en la estabilidad de los valores propios y la ausencia de cierto espectro singular. Estas perturbaciones son de diferente naturaleza, incluyendo rango finito y el caso singular. También se caracteriza y demuestra el fenómeno de resonancia desde el punto de vista dinámico, es decir, la existencia de estados que tienen larga vida y para los cuales la amplitud de sobrevivencia tiene un comportamiento casi exponencial. Además se incluye una discusión de acerca de varios problemas abiertos en el área.

Keywords

Resonances , spectral stability , finite rank perturbations

Mathematics Subject Classification:

81Q10 , 47A35 , 47B47

M. Angélica Astaburuaga Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile. https://orcid.org/0000-0002-8899-4251
Víctor H. Cortés Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile. https://orcid.org/0000-0002-5187-3130
Claudio Fernández Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile. https://orcid.org/0000-0002-7558-0180
Rafael Del Río IIMAS, Universidad Nacional Autónoma de México, México https://orcid.org/0000-0002-9842-6952

Pages: 505–522
Date Published: 2025-10-30
In Press

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