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Absolutely continuous spectrum and spectral transition for some continuous random operators

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Abstract

In this paper we consider two classes of random Hamiltonians on \(L^2({\mathbb R}^d)\): one that imitates the lattice case and the other a Schrödinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.

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References

  1. Aizenman M, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6 (1994) 1163–1182

    Article  MathSciNet  MATH  Google Scholar 

  2. Aizenman M and Molchanov S, Localization at large disorder and extreme energies, an elementary derivation, Commun. Math. Phys. 157 (1993) 245–274

    Article  MathSciNet  MATH  Google Scholar 

  3. Amrein W O, Boutet de Monvel Anne and Georgescu V, C 0-groups, commutators methods and spectral theory of N-body Hamiltonians, Progress in Mathematics 135 (Basel-Boston-Berlin: Birkhauser Verlag) (1996)

    Google Scholar 

  4. Auscher P, Weiss G and Wickerhauser M V, Local sine and cosine bases of Coiffman-Meyer and construction of smooth wavelets, in Wavelets: A tutorial in theory and applications (ed.) Charles K Chui (New York: Academic Press) (1992)

    Google Scholar 

  5. Combes J M, Hislop P and Mourre E, Spectral averaging, perturbation of singular spectra and localization, Trans. Am. Math. Soc. 348 (1996) 4883–4894

    Article  MathSciNet  MATH  Google Scholar 

  6. Daubechies I, Ten lectures on wavelets (Philadelphia: SIAM) (1992)

    Book  MATH  Google Scholar 

  7. Denisov S A and Kisalev A, Spectral Properties of Schrodinger Operators with Decaying Potentials, Spectral theory and mathematical physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proceedings of Symposia in Pure Mathematics, volume 76 (AMS, Providence) (2007)

  8. Demuth M, and Krishna M, Determining spectra in Quantum Theory, Progress in Mathematical Physics, vol. 44 (Boston: Birkhauser) (2005)

    Google Scholar 

  9. Froese R, Hasler D and Spitzer W, Absolutely continuous spectrum for the Anderson model on a tree: A geometric proof of Klein’s theorem, Commun. Math. Phys. 269(1) (2007) 239–257

    Article  MathSciNet  MATH  Google Scholar 

  10. Georgescu V and Gerard C, On the virial theorem in quantum mechanics, Commun. Math. Phys. 208 (1999) 275–281

    Article  MathSciNet  MATH  Google Scholar 

  11. Howland J, Random perturbations of singular spectra, Proc. Am. Math. Soc. 112 (1991) 1009–1011

    Article  MathSciNet  MATH  Google Scholar 

  12. Hislop P D, Kirsch W and Krishna M, Spectral and Dynamical properties of random models with nonlocal and singular interactions with P D Hislop and W Kirsch, Math. Nachrichten 278(6) (2005) 627–664

    Article  MathSciNet  MATH  Google Scholar 

  13. Jaksić V and Last Y, Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12 (2000) 1465–1503

    MathSciNet  MATH  Google Scholar 

  14. Jaksić V and Last Y, Surface states and spectra, Commun. Math. Phys. 218 (2001) 459–477

    Article  MATH  Google Scholar 

  15. Kalf H, The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators, Israel. J. Math. 20 (1975) 57–69

    Article  MathSciNet  MATH  Google Scholar 

  16. Klein A, Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998) 163–184

    Article  MathSciNet  MATH  Google Scholar 

  17. Kotani S, One-dimensional random Schrödinger operators and Herglotz functions, in: Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985) pp. 219–250 (Boston, MA: Academic Press) (1987)

    Google Scholar 

  18. Krishna M and Stollmann P, Direct integrals and spectral averaging, preprint (2010)

  19. Lemairé P and Meyer Y, Ondelettes et bases hilbertiennes, Rev. Iberoamericana 2 (1986) 1–18

    Google Scholar 

  20. Mourre E, Absence of singular continuous spectrum for some self-adjoint operators, Commun. Math. Phys. 78 (1981) 391–408

    Article  MathSciNet  MATH  Google Scholar 

  21. Perry P, Sigal I M and Simon B, Spectral analysis of N-body Schrödinger operators, Ann. Math. 114 (1981) 519–567

    Article  MathSciNet  MATH  Google Scholar 

  22. Reed M and Simon B, Methods of Modern Mathematical Physics IV: Analysis of Operators (New York: Academic Press) (1978)

    MATH  Google Scholar 

  23. Rodnianski I and Schlag W, Classical and quantum scattering for a class of long range random potentials, Int. Math. Res. Not. 5 (2003) 243–300

    Article  MathSciNet  Google Scholar 

  24. Safranov O, Absolutely continuous spectrum of a one-parameter family of Schrödinger operators, Preprint 11-82, MP_ARC (2011)

  25. Weidmann J, The virial theorem and its applications to the spectral theory of Schrödinger operators, Bull. Am. Math. Soc. 77 (1967) 452–456

    Article  MathSciNet  Google Scholar 

  26. Wojtaszczyk P, A Mathematical Introduction to Wavelets: London Mathematical Society Student Texts 37 (Cambridge University Press) (1997)

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  1. Institute of Mathematical Sciences, Taramani, Chennai 600 113, India

    M KRISHNA

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  1. M KRISHNA
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Correspondence to M KRISHNA.

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Dedicated to Barry Simon for his 65th birthday

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KRISHNA, M. Absolutely continuous spectrum and spectral transition for some continuous random operators. Proc Math Sci 122, 243–255 (2012). https://doi.org/10.1007/s12044-012-0069-4

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  • DOI: https://doi.org/10.1007/s12044-012-0069-4

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