Localization of the continuous Anderson Hamiltonian in 1-D

Abstract

We study the bottom of the spectrum of the Anderson Hamiltonian \({\mathcal {H}}_L := -\partial _x^2 + \xi \) on [0, L] driven by a white noise \(\xi \) and endowed with either Dirichlet or Neumann boundary conditions. We show that, as \(L\rightarrow \infty \), the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on \(\mathbf{R}\) with intensity \(e^x dx\), and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.

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Acknowledgements

We thank the anonymous referees for their careful reading of the paper and their helpful remarks. LD thanks Romain Allez and Benedek Valkó for useful discussions. The work of LD is supported by the project MALIN ANR-16-CE93-0003. CL is grateful to Julien Reygner for several useful comments on a preliminary version of this paper. The work of CL is supported by the project SINGULAR ANR-16-CE40-0020-01.

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Correspondence to Cyril Labbé.

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Dumaz, L., Labbé, C. Localization of the continuous Anderson Hamiltonian in 1-D. Probab. Theory Relat. Fields 176, 353–419 (2020). https://doi.org/10.1007/s00440-019-00920-6

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Keywords

  • Anderson Hamiltonian
  • Hill’s operator
  • Localization
  • Riccati transform
  • Diffusion

Mathematics Subject Classification

  • Primary 60H25
  • 60J60
  • Secondary 35P20