Localization crossover for the continuous Anderson Hamiltonian in 1-d

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian H-L, with white noise potential, on a segment whose size L is sent to infinity. We zoom around energy levels E either of order 1 (Bulk regime) or of order 1 << E << L (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper (Dumaz and Labbe in Ann. Probab. 51(3):805-839, 2023), this identifies completely the transition between the localized and delocalized phases of the spectrum of H-L. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.