Probabilistic Weyl Laws for Quantized Tori

“…A series of papers by W. Bordeaux-Montrieux, M. Hager and J. Sjöstrand [11,2,10,3,13,23,24,22] established a probabilistic Weyl law in the interior of the pseudospectrum for a large class of elliptic (pseudo-)differential operators subject to small random perturbations in the semiclassical or high energy limit. Furthermore, a similar result has been obtained by T. Christiansen and M. Zworski for certain randomly perturbed Toeplitz operators in [4].…”

Section: Introductionsupporting

confidence: 81%

Two Point Eigenvalue Correlation for a Class of Non-Selfadjoint Operators Under Random Perturbations

Vogel

2016

Commun. Math. Phys.

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Abstract. We consider a non-selfadjoint h-differential model operator P h in the semiclassical limit (h → 0) subject to random perturbations with a small coupling constant δ. Assume that e − 1Ch < δ h κ for constants C, κ > 0 suitably large. Let Σ be the closure of the range of the principal symbol.We study the 2-point intensity measure of the random point process of eigenvalues of the randomly perturbed operator P δ h and prove an h-asymptotic formula for the average 2-point density of eigenvalues. With this we show that two eigenvalues of P δ h in the interior of Σ exhibit close range repulsion and long range decoupling.Résumé Nous considérons un opérateur différentiel non-autoadjoint P h dans la limite semiclassique (h → 0) soumis à de petites perturbations aléatoires. De plus, nous imposons que la constant de couplage δ vérifie e − 1Ch < δ h κ pour certaines constantes C, κ > 0 choisies assez grandes. Soit Σ l'adhérence de l'image du symbole principal de P h .Dans cet article, nous donnons une formule h-asymptotique pour la 2-points densité des valeurs propres en étudiant la mesure de comptage aléatoire des valeurs propres à l'intérieur de Σ. En étudiant cette densité, nous prouvons que deux valeurs propres sont répulsives à distance courte et indépendantes à long distance.

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“…It is therefore a natural question to study the spectra of such operators subject to small random perturbations. Recently, there has been a mounting interest in the spectral properties of elliptic non-normal (pseudo-)differential operators with small random perturbations, see for example [2,10,12,17,22,4]. An interesting, perhaps surprising, result is that by adding a small random perturbation, we can obtain a probabilistic Weyl law for the eigenvalues for a large class of such operators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Interior Eigenvalue Density of Jordan Matrices with Random Perturbations

Sjöstrand

Vogel

2017

Trends in Mathematics

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Abstract. We study the eigenvalue distribution of a large Jordan block subject to a small random Gaussian perturbation. A result by E.B. Davies and M. Hager shows that as the dimension of the matrix gets large, with probability close to 1, most of the eigenvalues are close to a circle.We study the expected eigenvalue density of the perturbed Jordan block in the interior of that circle and give a precise asymptotic description.Résumé. Nous étudions la distribution de valeurs propres d'un grand bloc de Jordan soumis à une petite perturbation gaussienne aléatoire. Un résultat de E.B. Davies et M. Hager montre que quand la dimension de la matrice devient grande, alors avec probabilité proche de 1, la plupart des valeurs propres sont proches d'un cercle.Nous étudions la répartitions moyenne des valeurs propres à l'intérieur de ce cercle et nous en donnons une description asymptotique précise.

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Probabilistic Weyl Laws for Quantized Tori

Cited by 16 publications

References 18 publications

Spectral statistics of non-selfadjoint operators subject to small random perturbations

Spectral statistics of non-selfadjoint operators subject to small random perturbations

Two Point Eigenvalue Correlation for a Class of Non-Selfadjoint Operators Under Random Perturbations

Interior Eigenvalue Density of Jordan Matrices with Random Perturbations

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