Skip to main content

Advertisement

SpringerLink
Log in

Localization of the continuous Anderson Hamiltonian in 1-D

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Allez, R., Chouk, K.: The continuous Anderson Hamiltonian in dimension two. ArXiv e-prints (2015). arXiv:1511.02718

  2. Allez, R., Dumaz, L.: From sine kernel to Poisson statistics. Electron. J. Probab. 19(114), 25 (2014). https://doi.org/10.1214/EJP.v19-3742

    Article  MathSciNet  MATH  Google Scholar 

  3. Allez, R., Dumaz, L.: Tracy–Widom at high temperature. J. Stat. Phys. 156(6), 1146–1183 (2014). https://doi.org/10.1007/s10955-014-1058-z

    Article  MathSciNet  MATH  Google Scholar 

  4. Allez, R., Dumaz, L.: Random matrices in non-confining potentials. J. Stat. Phys. 160(3), 681–714 (2015). https://doi.org/10.1007/s10955-015-1258-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492–1505 (1958). https://doi.org/10.1103/PhysRev.109.1492

    Article  Google Scholar 

  6. Biskup, M., König, W.: Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails. Commun. Math. Phys. 341(1), 179–218 (2016). https://doi.org/10.1007/s00220-015-2430-9

    Article  MATH  Google Scholar 

  7. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae. Probability and Its Applications, 2nd edn. Birkhäuser Verlag, Basel (2002)

    Book  Google Scholar 

  8. Bloemendal, A., Virág, B.: Limits of spiked random matrices II. Ann. Probab. 44(4), 2726–2769 (2016). https://doi.org/10.1214/15-AOP1033

    Article  MathSciNet  MATH  Google Scholar 

  9. Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108(1), 41–66 (1987)

    Article  MathSciNet  Google Scholar 

  10. Cambronero, S., McKean, H.P.: The ground state Eigenvalue of Hill’s equation with white noise potential. Commun. Pure Appl. Math. 52(10), 1277–1294 (1999). https://doi.org/10.1002/(SICI)1097-0312(199910)52:10<1277::AID-CPA5>3.0.CO;2-L

    Article  MathSciNet  Google Scholar 

  11. Cambronero, S., Rider, B., Ramírez, J.: On the shape of the ground state eigenvalue density of a random Hill’s equation. Commun. Pure Appl. Math. 59(7), 935–976 (2006). https://doi.org/10.1002/cpa.20104

    Article  MathSciNet  MATH  Google Scholar 

  12. Dumaz, L., Virág, B.: The right tail exponent of the Tracy–Widom \(\beta \) distribution. Ann. Inst. Henri Poincaré Probab. Stat. 49(4), 915–933 (2013). https://doi.org/10.1214/11-AIHP475

    Article  MathSciNet  MATH  Google Scholar 

  13. Frisch, H.L., Lloyd, S.P.: Electron levels in a one-dimensional random lattice. Phys. Rev. 120, 1175–1189 (1960). https://doi.org/10.1103/PhysRev.120.1175

    Article  MATH  Google Scholar 

  14. Fukushima, M., Nakao, S.: On spectra of the Schrödinger operator with a white Gaussian noise potential. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37(3), 267–274 (1976/77)

  15. Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6, 75 (2015). arXiv:1210.2684. https://doi.org/10.1017/fmp.2015.2

  16. Goldsheid, I.J., Molcanov, S.A., Pastur, L.A.: A random homogeneous Schrödinger operator has a pure point spectrum. Funkcional. Anal. i Priložen. 11(1), 1–10 (1977). 96

    Article  MathSciNet  Google Scholar 

  17. Gubinelli, M., Ugurcan, B., Zachhuber, I.: Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions. ArXiv e-prints (2018). arXiv:1807.06825

  18. Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014). arXiv:1303.5113. https://doi.org/10.1007/s00222-014-0505-4

    Article  MathSciNet  Google Scholar 

  19. Halperin, B.I.: Green’s functions for a particle in a one-dimensional random potential. Phys. Rev. (2) 139, A104–A117 (1965)

    Article  MathSciNet  Google Scholar 

  20. König, W.: The Parabolic Anderson Model. Pathways in Mathematics. Random Walk in Random Potential. Birkhäuser/Springer, Cham (2016)

    Book  Google Scholar 

  21. Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  22. Kirsch, W.: An invitation to random Schrödinger operators. In Random Schrödinger operators, vol. 25 of Panor. Synthèses, 1–119. Soc. Math. France, Paris, 2008. With an appendix by Frédéric Klopp

  23. Killip, R., Nakano, F.: Eigenfunction statistics in the localized Anderson model. Ann. Henri Poincaré 8(1), 27–36 (2007). https://doi.org/10.1007/s00023-006-0298-0

    Article  MathSciNet  MATH  Google Scholar 

  24. Killip, R., Stoiciu, M.: Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146(3), 361–399 (2009). https://doi.org/10.1215/00127094-2009-001

    Article  MathSciNet  MATH  Google Scholar 

  25. Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78(2), 201–246 (1980/81)

    Article  MathSciNet  Google Scholar 

  26. Kritchevski, E., Valkó, B., Virág, B.: The scaling limit of the critical one-dimensional random Schrödinger operator. Commun. Math. Phys. 314(3), 775–806 (2012). https://doi.org/10.1007/s00220-012-1537-5

    Article  MATH  Google Scholar 

  27. Labbé, C.: The continuous Anderson Hamiltonian in \(d\le 3\). ArXiv e-prints (2018). arXiv:1809.03718

  28. McKean, H.P.: A limit law for the ground state of Hill’s equation. J. Stat. Phys. 74(5–6), 1227–1232 (1994). https://doi.org/10.1007/BF02188225

    Article  MathSciNet  MATH  Google Scholar 

  29. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177(3), 709–725 (1996)

    Article  MathSciNet  Google Scholar 

  30. Molčanov, S.A: The local structure of the spectrum of the one-dimensional Schrödinger operator. Commun. Math. Phys. 78(3), 429–446 (1980/81)

    Article  Google Scholar 

  31. Otto, F., Weber, H., Westdickenberg, M.G.: Invariant measure of the stochastic Allen–Cahn equation: the regime of small noise and large system size. Electron. J. Probab. 19(23), 76 (2014). https://doi.org/10.1214/EJP.v19-2813

    Article  MathSciNet  MATH  Google Scholar 

  32. Ramírez, J.A., Rider, B.: Diffusion at the random matrix hard edge. Commun. Math. Phys. 288(3), 887–906 (2009). https://doi.org/10.1007/s00220-008-0712-1

    Article  MathSciNet  MATH  Google Scholar 

  33. Ramírez, J.A., Rider, B., Virág, B.: Beta ensembles, stochastic airy spectrum, and a diffusion. J. Am. Math. Soc. 24(4), 919–944 (2011). https://doi.org/10.1090/S0894-0347-2011-00703-0

    Article  MathSciNet  MATH  Google Scholar 

  34. Rifkind, B., Virág, B.: Eigenvectors of the critical 1-dimensional random Schrödinger operator. ArXiv e-prints (2016). arXiv:1605.00118

  35. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. Springer, Berlin (1999)

    Book  Google Scholar 

  36. Texier, C.: Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder. J. Phys. A 33(35), 6095–6128 (2000). https://doi.org/10.1088/0305-4470/33/35/303

    Article  MathSciNet  MATH  Google Scholar 

  37. Valkó, B., Virág, B.: Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177(3), 463–508 (2009). https://doi.org/10.1007/s00222-009-0180-z

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. UMR 7534, CNRS, CEREMADE, Université Paris-Dauphine, PSL Research University, 75016, Paris, France

    Laure Dumaz & Cyril Labbé

Authors
  1. Laure Dumaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Labbé
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyril Labbé.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-019-00920-6

Keywords

Mathematics Subject Classification

Search

Navigation