Laure Dumaz

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

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.