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Localization of the continuous Anderson Hamiltonian in 1-D

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1711.04700.pdf (1.713Mb)
Date
2019
Dewey
Probabilités et mathématiques appliquées
Sujet
Anderson Hamiltonian; Hill’s operator; Localization; Riccati transform; Diffusion
Journal issue
Probability Theory and Related Fields
Volume
176
Publication date
2019
Article pages
353–419
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s00440-019-00920-6
URI
https://basepub.dauphine.fr/handle/123456789/19673
Collections
  • CEREMADE : Publications
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Author
Dumaz, Laure
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Labbé, Cyril
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
We study the bottom of the spectrum of the Anderson Hamiltonian HL:=−∂2x+ξ on [0, L] driven by a white noise ξ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L→∞, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity exdx, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.

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