Résumé (EN)

We introduce a random differential operator, that we call the CSτ operator, whose spectrum is given by the Schτ point process introduced by Kritchevski, Valk\'o and Vir\'ag (2012) and whose eigenvectors match with the description provided by Rifkind and Vir\'ag (2018). This operator acts on R2-valued functions from the interval [0,1] and takes the form:2(0∂t−∂t0)+τ−−√⎛⎝dB+12√dW112√dW212√dW2dB−12√dW1⎞⎠,where dB, dW1 and dW2 are independent white noises. Then, we investigate the high part of the spectrum of the Anderson Hamiltonian HL:=−∂2t+dB on the segment [0,L] with white noise potential dB, when L→∞. We show that the operator HL, recentred around energy levels E∼L/τ and unitarily transformed, converges in law as L→∞ to CSτ in an appropriate sense. This allows to answer a conjecture of Rifkind and Vir\'ag (2018) on the behavior of the eigenvectors of HL. Our approach also explains how such an operator arises in the limit of HL. Finally we show that at higher energy levels, the Anderson Hamiltonian matches (asymptotically in L) with the unperturbed Laplacian −∂2t. In a companion paper, it is shown that at energy levels much smaller than L, the spectrum is localized with Poisson statistics: the present paper therefore identifies the delocalized phase of the Anderson Hamiltonian.