Localization of the continuous Anderson Hamiltonian in 1-D

Abstract

We study the bottom of the spectrum of the Anderson Hamiltonian HL := −∂ 2 x + 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 e x dx, 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.

2D random magnetic Laplacian with white noise magnetic field

“…Similarily, would it be possible to find an effective "classical" equation describing the semiclassical limit in the case of a singular potential V = ξ or B = ξ ? See the book [12] from Helffer for details on the semiclassical limit and the work [6] by Dumaz and Labbé for a very precise description of the asymptotics in one dimension for the Anderson Hamiltonian.…”

Operator level hard-to-soft transition for β-ensembles

“…]. (We refer to Lemma 5.7. of [5] for additional details for this argument.) By Markov's inequality, we get…”

Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise

“…The first mathematically rigorous study of this object appeared in [17]. Following this, there have been several investigations of A's spectral properties [5,6,31], culminating in a recent article of Dumaz and Labbé [13], which provides a comprehensive description of eigenfunction localization and eigenvalue Poisson statistics in the case where A acts on I = (0, L) for large L.…”