Poisson statistics for eigenvalues of continuum random Schrödinger operators

Abstract

We show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the local- ization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point pro- cess with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum An- derson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model. In this article we show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the localiza- tion region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues in that region. Local fluctuations of eigenvalues of random operators is believed to distinguish between localized and delocalized regimes, indicating an Anderson metal-insulator transition. Exponential decay of eigenfunctions implies that disjoint regions of space are uncorrelated and create almost independent eigenvalues, and thus absence of energy levels repulsion, which is mathematically translated in terms of a Poisson point process. On the other hand, extended states imply that distant regions have mutual influence, and thus create some repulsion between energy levels. Local fluctuations of eigenvalues have been studied within the context of random matrix theory, in particular Wigner matrices and GUE matrices, cf. (B, DiPS, ESY1, ESY2, J1, J2, SS) and references therein. It is challenging to understand random hermitian band matrices from the perspective of their eigenvalues fluctuations, by proving a transition between Poisson statistics and a semi-circle law for the density of states (a signature of energy levels repulsion), and relate this to the (discrete) Anderson model, cf. (B, DiPS). CMV matrices are another class of random matrices for which Poisson statistics and a transition to energy levels repulsion have been proved (KS, St1, St2). For random Schrodinger operators, Poisson statistics for eigenvalues was first proved by Molchanov (Mo2) for the same one-dimensional continuum random Schro- dinger operator for which Anderson localization was first rigorously established