### Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization

Probability Seminar

23rd June 2021, 4:00 pm – 5:00 pm

online, online

We consider the 1-dimensional continuous Schrodinger

operator - d^2/d^x^2 + B’(x) on an interval of size L where the

potential B’ is a white noise. We study the entire spectrum of this

operator in the large L limit. We prove the joint convergence of the

eigenvalues and of the eigenvectors and describe the limiting shape of

the eigenvectors for all energies. When the energy is much smaller

than L, we find that we are in the localized phase and the eigenvalues

are distributed as a Poisson point process. The transition towards

delocalization holds for large eigenvalues of order L. In this regime,

we show the convergence at the level of operators. The limiting

operator is acting on R^2-valued functions and is of the form ``J

\partial_t + 2*2 noise matrix'' (where J is the matrix ((0, -1)(1,

0))), a form which already appeared as a conjecture by Edelman Sutton

(2006) for limiting random matrices. Joint works with Cyril Labbé.

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