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Localization of the continuous Anderson Hamiltonian in 1-D

Dumaz, Laure; Labbé, Cyril (2019), Localization of the continuous Anderson Hamiltonian in 1-D, Probability Theory and Related Fields, 176, p. 353–419. 10.1007/s00440-019-00920-6

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Type
Article accepté pour publication ou publié
Date
2019
Journal name
Probability Theory and Related Fields
Volume
176
Publisher
Springer
Pages
353–419
Publication identifier
10.1007/s00440-019-00920-6
Metadata
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Author(s)
Dumaz, Laure
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Labbé, Cyril
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
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.
Subjects / Keywords
Anderson Hamiltonian; Hill’s operator; Localization; Riccati transform; Diffusion

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