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

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Laure Dumaz
École Normale supérieure

We consider the continuous Schrödinger operator - d^2/d^x^2 + B’(x) on the interval [0,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 in the delocalized phase 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 appearing as a conjecture by Edelman Sutton (2006) for limiting random matrices. Joint works with Cyril Labbé.