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The delocalized phase of the Anderson Hamiltonian in 1-d

hal.structure.identifierDépartement de Mathématiques et Applications - ENS Paris [DMA]
dc.contributor.authorDumaz, Laure
HAL ID: 739617
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorLabbé, Cyril
HAL ID: 9675
dc.date.accessioned2022-02-28T10:51:47Z
dc.date.available2022-02-28T10:51:47Z
dc.date.issued2021
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/22794
dc.language.isoenen
dc.subjectAnderson Hamiltonianen
dc.subjectHill’s operatoren
dc.subjectCanonical systemsen
dc.subjectDirac operatoren
dc.subjectDelocalizationen
dc.subjectStrong resolvent convergenceen
dc.subjectDiffusionen
dc.subjectSchen
dc.subject.ddc519en
dc.titleThe delocalized phase of the Anderson Hamiltonian in 1-den
dc.typeDocument de travail / Working paper
dc.description.abstractenWe introduce a random differential operator, that we call the CSτ operator, whose spectrum is given by the Schτ point process introduced by Kritchevski, Valk\'o and Vir\'ag (2012) and whose eigenvectors match with the description provided by Rifkind and Vir\'ag (2018). This operator acts on R2-valued functions from the interval [0,1] and takes the form:2(0∂t−∂t0)+τ−−√⎛⎝dB+12√dW112√dW212√dW2dB−12√dW1⎞⎠,where dB, dW1 and dW2 are independent white noises. Then, we investigate the high part of the spectrum of the Anderson Hamiltonian HL:=−∂2t+dB on the segment [0,L] with white noise potential dB, when L→∞. We show that the operator HL, recentred around energy levels E∼L/τ and unitarily transformed, converges in law as L→∞ to CSτ in an appropriate sense. This allows to answer a conjecture of Rifkind and Vir\'ag (2018) on the behavior of the eigenvectors of HL. Our approach also explains how such an operator arises in the limit of HL. Finally we show that at higher energy levels, the Anderson Hamiltonian matches (asymptotically in L) with the unperturbed Laplacian −∂2t. In a companion paper, it is shown that at energy levels much smaller than L, the spectrum is localized with Poisson statistics: the present paper therefore identifies the delocalized phase of the Anderson Hamiltonian.en
dc.publisher.cityParisen
dc.identifier.citationpages31en
dc.relation.ispartofseriestitleCahier de recherche CEREMADE, Université Paris Dauphine-PSLen
dc.identifier.urlsitehttps://hal.archives-ouvertes.fr/hal-03370032en
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.identifier.citationdate2021
dc.description.ssrncandidatenon
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.date.updated2022-02-28T10:44:23Z
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