Another look at qualitative properties of eigenvalues using effective Hamiltonians

Idriss Mazari-Fouquer

doi:10.5802/crmath.747

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Another look at qualitative properties of eigenvalues using effective Hamiltonians
[Un autre point de vue sur les propriétés qualitatives des valeurs propres via les hamiltoniens effectifs]

Idriss Mazari-Fouquer ¹

¹ CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 677-693.

Abstract (VO)
Résumé

The goal of this paper is to review several qualitative properties of well-known eigenvalue problems using a different perspective based on the theory of effective Hamiltonians, working exclusively on the Hopf–Cole transform of the equation. We revisit some monotonicity results as well as the derivation of several scaling limits by means of the Donsker–Varadhan formula, and we point out several differences between the case of quadratic Hamiltonians and non-quadratic ones.

L’objectif de ce papier est de passer en revue plusieurs propriétés qualitatives de problèmes spectraux bien connus en utilisant une autre perspective, celles de hamiltoniens effectifs, en travaillant uniquement sur la transformée de Hopf–Cole de l’équation. Nous revisitons certains résultats de monotonie et de comportements asymptotiques en utilisant la formule de Donsker–Varadhan, et nous mettons en lumière plusieurs différences fondamentales entre les hamiltoniens quadratiques, et ceux qui ne le sont pas.

Reçu le : 2025-03-03
Révisé le : 2025-04-01
Accepté le : 2025-04-01
Publié le : 2025-06-23

DOI : 10.5802/crmath.747

Keywords: Effective Hamiltonian, principal eigenvalue, spectral optimisation
Mots-clés : Hamiltonien effectif, valeur propre principale, optimisation spectrale

Affiliations des auteurs :

Idriss Mazari-Fouquer ¹

¹ CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Université PSL, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Another look at qualitative properties of eigenvalues using effective {Hamiltonians}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {677--693},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.747},
     language = {en},
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Idriss Mazari-Fouquer. Another look at qualitative properties of eigenvalues using effective Hamiltonians. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 677-693. doi : 10.5802/crmath.747. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.747/

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