Exponential localization for multi-dimensional Schrödinger operator with random point potential

Reviews in Mathematical Physics

Volumn 9, Issue 4, 1997, Pages 425-451

  Boutet De Monvel, Anne ^a,b   Grinshpun, Vadim ^b  

^a INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

^b B VERKIN INSTITUTE FOR LOW TEMPERATURE PHYSICS AND ENGINEERING   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031138482     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/s0129055x97000178     Document Type: Article

References (35)

1. S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, Berlin, Heidelberg, 1988.
2. S. Albeverio, R. Hoegh-Krohn, W. Kirsch and F. Martinelli, "The spectrum of the three-dimensional Kronig-Penney model with random point defects", Adv. Appl. Math. 3 (1982) 435-440.
3. M. Aizenman and S. Molchanov, "Localization at large disorder and at extreme energies: an elementary derivation", Commun. Math. Phys. 157 (1993) 245-279.
4. J. Berezanskii, Expansion in Eigenfunctions of Self-adjoint Operators, Transl. Math. Monographs 17, American Mathematical Society, Providence, R.I., 1968.
5. A. Boutet de Monvel and V. Grinshpun, "Expansion in generalized eigenfunctions of point interactions hamiltonians", in preparation.
6. R. Carmona, A. Klein and F. Martinelli, "Anderson localization for Bernoulli and other singular potentials", Commun. Math. Phys. 108 (1987) 41-66.
7. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
8. H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1987.
9. J.-M. Combes and P. D. Hislop, "Localization for some continuous random Hamiltonians in d-dimensions", J. Funct. Anal. 124 (1994) 149-180.
10. J.-M. Combes, P. D. Hislop and E. Mourre, "Spectral averaging, perturbation of singular spectra, and localization", Trans. Amer. Math. Soc. 348 (1996) 4883-4894.
11. F. Delyon, Y. Levi and B. Souillard, "Anderson localization for multidimensional systems at large disorder or large energy", Commun. Math. Phys. 100 (1985) 463-470.
12. F. Delyon, B. Simon and B. Souillard, "From power pure point to continuous spectrum in disordered systems", Ann. Inst. H.Poincaré, Phys. Théor. 42 (1985) 283-309.
13. H. von Dreifus: "On the effects of randomness in ferromagnetic models and Schrödinger operators", NYU Ph.D. Thesis, 1987.
14. H. von Dreifus and A. Klein, "A new proof of localization in the Anderson tight binding model", Commun. Math. Phys. 124 (1989) 285-299.
15. J. Fröhlich and T. Spencer, "Absence of diffusion in the Anderson tight binding model for large disorder or low energy", Commun. Math. Phys. 88 (1983) 151-184.
16. J. Fröhlich, F. Martinelli, E. Scoppola and T. Spencer, "Constructive proof of localization in the Anderson tight binding model", Commun. Math. Phys. 101 (1985) 21-46.
17. I. Goldsheid, S. Molchanov and L. Pastur, "A pure point spectrum of the stochastic one-dimensional Schrödinger operator", Funct. Anal. Appl. 11 (1977) 1-10.
18. d)", Reports Ukrainian Acad. of Sci. (8) (1992) 18-21; (5) (1993) 26-29 (in Russian).
19. d)", Reports Ukrainian Acad. of Sci. (8) (1992) 18-21; (5) (1993) 26-29 (in Russian).
20. V. Grinshpun, "Point spectrum of finite-difference multi-dimensional random operators", PhD Thesis, University Paris 7, 1994.
21. V. Grinshpun, "Localization for random potentials supported on a subspace", Lett. Math. Phys. 34 (2) (1995) 103-117.
22. W. Kirsch and F. Martinelli, "On the spectrum of Schrödinger operators with a random potential", J. Phys. A15 (1982) 2139-2156.
23. W. Kirsch, "Random Schrödinger operators and the density of states", Stochastic Aspects of Classical and Quantum Systems, eds. S. Albeverio, P. Combe, M. Sirugue-Collin, Marseille 1983, Lecture Notes in Math. 1109, Springer-Verlag, Berlin, Heidelberg, New York, 1985, 68-102.
24. A. Klein, "Localization in the Anderson model with long range hopping", Braz. J. Phys. 23 (1993) 363-371.
25. V. Korotkov, "Integral operators with Carleman kernels", Dokl. Akad. Nauk SSSR 165 (1965) 748-757.
26. S. Kotani and B. Simon, "Localization in general one-dimensional random systems: continuum Schrödinger operators. II", Commun. Math. Phys. 112 (1987) 103-119.
27. V. Kovalenko and Yu. Semenov, "Some problems on expansions in generalized eigenfunctions of the Schrödinger operator with strongly singular potentials", Russian Math. Surveys 33(4) (1978) 119-157.
28. v)", Commun. Math. Phys. 93 (1984) 197-217.
29. L. Pastur and A. Figotin, Spectra of Random and Almost Periodic Operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, Heidelberg, New-York, 1992.
30. B. Simon, "Schrödinger semigroups", Bull. Amer. Math. Soc. 7 (3) (1982) 447-527.
31. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, San Francisco, London, 1979.
32. B. Simon and T. Wolff, "Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians", Commun. Pure Appl. Math. 39 (1986) 75-90.
33. J. Sch'noll, "On the behaviour of the Schrödinger equation", Mat. Sb. 42 (1957) 273-286 (in Russian).
34. T. Spencer, "Localization for random and quasi-periodic potentials", J. Stat. Phys. 51 (1988) 1009-1019.
35. F. Wegner, "Bounds on the density of states in disordered systems", Z. Phys. B44 (1981) 9-15.