Équilibre instable en régime semi-classique: i—concentration microlocale

Verdìère, Yves Colin de; Parisse, Bernard

doi:10.1080/03605309408821063

Cited by 79 publications

(40 citation statements)

References 23 publications

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“…The following result as stated can be read off from [Chr07,Chr11], and has also been studied in slightly different contexts in [CdVP94a,CdVP94b] and [BZ04], amongst many others.…”

Section: 2mentioning

confidence: 99%

Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

Christianson

2015

Annales De l'Institut Fourier

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Abstract. We prove a strong conditional unique continuation estimate for irreducible quasimodes in rotationally invariant neighbourhoods on compact surfaces of revolution. The estimate states that Laplace quasimodes which cannot be decomposed as a sum of other quasimodes have L 2 mass bounded below by Cǫλ −1−ǫ for any ǫ > 0 on any open rotationally invariant neighbourhood which meets the semiclassical wavefront set of the quasimode. For an analytic manifold, we conclude the same estimate with a lower bound of C δ λ −1+δ for some fixed δ > 0.

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“…The following result as stated can be read off from [Chr07,Chr11], and has also been studied in slightly different contexts in [CdVP94a,CdVP94b] and [BZ04], amongst many others.…”

Section: 2mentioning

confidence: 99%

Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

Christianson

2015

Annales De l'Institut Fourier

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“…En effet, pour un hamiltonien p : M → R tel que 0 soit valeur critique de p, et tel que les fibres dans un voisinage de 0 soient compactes et connexes et ne contiennent qu'un unique point critique non-dégénéré de type hyperbolique : la fibre Λ 0 := p −1 (0) est alors un "huit" et le feuilletage dans un voisinage de la fibre singulière Λ 0 est difféomorphe à celui du double puits. [10] et [11] les deux auteurs donnent une condition nécessaire et suffisante pour trouver le spectre semi-classique dans un compact de diamètre h centré autour de l'origine de l'opérateur linéaire :…”

Section: Olivier Labléeunclassified

“…On résume [10], [11] et une bonne partie de [12]. La preuve de la formule se décompose en plusieurs grandes étapes.…”

Section: Les Grandes éTapes De La Preuveunclassified

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Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique

Lablée¹

2008

Séminaire De Théorie Spectrale Et Géométrie

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Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type

Ngọc

2000

Comm. Pure Appl. Math.

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We present a detailed study, in the semi-classical regime h → 0, of microlocal properties of systems of two commuting h-pseudo-differential operators P 1 (h), P 2 (h) such that the joint principal symbol p = (p 1 , p 2 ) has a special kind of singularity called a focus-focus singularity. Typical examples include the quantum spherical pendulum or the quantum Champagne bottle.In the spirit of Colin de Verdière and Parisse [11,12,13], we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type.These conditions are used to give a precise description of the joint spectrum of such systems, including the phenomenon of quantum monodromy and different formulations of the counting function for the joint eigenvalues close to the singularity, in which a logarithm of the semi-classical constant h appears. Thanks to numerical computations done by M.S. Child for the case of the Champagne bottle, we are able to accurately illustrate our statements.

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Équilibre instable en régime semi-classique: i—concentration microlocale

Cited by 79 publications

References 23 publications

Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

Unique Continuation for Quasimodes on Surfaces of Revolution: Rotationally invariant Neighbourhoods

Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique

Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type

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