Some aspects of the Anderson Hamiltonian with white noise

In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « - Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are localized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.