May 2023 The delocalized phase of the Anderson Hamiltonian in 1-D
Laure Dumaz, Cyril Labbé
Author Affiliations +
Ann. Probab. 51(3): 805-839 (May 2023). DOI: 10.1214/22-AOP1591


We introduce a random differential operator that we call CSτ operator, whose spectrum is given by the Schτ point process introduced by Kritchevski, Valkó and Virág (Comm. Math Phys. (2012) 314 775–806) and whose eigenvectors match with the description provided by Rifkind and Virág (Geom. Funct. Anal. (2018) 28 1394–1419). This operator acts on R2-valued functions from the interval [0,1] and takes the form


where dB, dW1 and dW2 are independent white noises. Then we investigate the high part of the spectrum of the Anderson Hamiltonian HL:=t2+dB on the segment [0,L] with white noise potential dB, when L. We show that the operator HL, recentred around energy levels EL/τ and unitarily transformed, converges in law as L to CSτ in an appropriate sense. This allows us to answer a conjecture of Rifkind and Virág 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 t2. 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.

Funding Statement

The work of CL is supported by the project SINGULAR ANR-16-CE40-0020-01. CL was part-time affiliated to École Normale Supérieure, DMA in 2020–2021.


The authors thank the referees for their useful comments on the first version of this paper. Special thanks are due to one anonymous referee for pointing out the link between the operator Schτ, defined by Valkó and Virág, and the operator constructed in this paper which led to Theorem 1.4 and the new Section 2.3. The authors also thank Gaultier Lambert, Benedek Valkó and Bálint Virág for their constructive remarks and corrections on the first version of this paper.


Download Citation

Laure Dumaz. Cyril Labbé. "The delocalized phase of the Anderson Hamiltonian in 1-D." Ann. Probab. 51 (3) 805 - 839, May 2023.


Received: 1 April 2021; Revised: 1 February 2022; Published: May 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583056
zbMATH: 07690049
Digital Object Identifier: 10.1214/22-AOP1591

Primary: 60H25 , 60J60
Secondary: 60B20

Keywords: Anderson Hamiltonian , canonical systems , delocalization , diffusion , Dirac operator , Hill’s operator , SCH , strong resolvent convergence

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 3 • May 2023
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