Internal lifshits tails for random perturbations of periodic schrödinger operators

Duke Mathematical Journal

Volumn 98, Issue 2, 1999, Pages 335-396

  Klopp, Frédéric ^a  

^a UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033148063     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-99-09810-1     Document Type: Article

References (41)

1. S. Agmon, “On positive solutions of elliptic equations with periodic coefficients in R", spectral results and extensions to elliptic operators on Riemannian manifolds” in Differential Equations (Birmingham Ala., 1983), ed. I. Knowles and R. Lewis, North-Holland Math. Stud. 92, North-Holland, Amsterdam, 1984, 7-17.
2. J.-M. Barbaroux, J.-M. Combes, and P. Hislop, Localization near band edges for random Schrodinger operators, Helv. Phys. Acta. 70 (1997), 16-43.
3. M. Sh. Birman, “Discrete spectrum in a gap of perturbed periodic operator at large coupling constants” in Rigorous Results in Quantum Dynamics (Liblice, 1990), ed. J. Dittrich and P. Exner, World Scientific Publishing, River Edge, N.J., 1991, 16-24.
4. R. Carmona, “Random Schrodinger operators” in Ecole d'été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math. 1180, Springer-Verlag, Berlin, 1986, 1-124.
5. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrodinger Operators with Applica tion to Quantum Mechanics and Global Geometry, Texts Monographs Phys., Springer-Verlag, Berlin, 1987.
6. F. Daumer, Equation de Schrodinger avec champ électrique périodique et champ magnétique constant dans l'approximation du tight binding, Comm. Partial Differential Equations 18 (1993), 1021-1041.
7. Y. C. deVerdière, “Sur les singularités de Van Hove génériques” in Analyse globale et physique mathématique: Colloquium en l'honneur d'E. Combet, Lyon, 1989, Mém. Soc. Math. France (N.S.) 46, (1991), 99-110.
8. M. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.
9. B. Helffer and J. Sjostrand, On diamagnetism and de Haas-van Alphen effect, Ann. Inst. H. Poincaré Phys. Théor. 52 (1990), 303-375.
10. D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. of Math. (2) 121 (1985), 463-494.
11. T. Kato, Perturbation Theory for Linear Operators, 2d ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, 1980.
12. W. Kirsch, “Random Schrodinger operators: Acourse” in Schrodinger Operators (S0nderborg, 1988), ed. A. Jensen and H. Holden, Lecture Notes in Phys. 345, Springer-Verlag, Berlin, 1989, 264-370.
13. W. Kirsch and F. Martinelli, On the spectrum of Schrodinger operators with a random potential, Comm. Math. Phys. 85 (1982), 329-350.
14. W. Kirsch and F. Martinelli, Large deviations and Lifshitz singularities of the integrated density of states of random Hamiltonians, Comm. Math. Phys. 89 (1983), 27-40.
15. W. Kirsch and B. Simon, Lifshits tails for the Anderson model, J. Statist. Phys. 38 (1985), 65-76.
16. W. Kirsch, P. Stollman, and G. Stolz, Localization for random perturbations of periodic Schrodinger operators, Random Oper. Stochastic Equations 6 (1998), 241-268.
17. F. Klopp, Etude semi-classique d'une perturbation d 'un opérateur de Schrodinger périodique, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), 459-509.
18. F. Klopp, An asymptotic expansion for the density of states of a random Schrodinger operator with Bernoulli disorder, Random Oper. Stochastic Equations 3 (1995), 315-331.
19. F. Klopp, Localization for some continuous random Schrodinger operators, Comm. Math. Phys. 167 (1995), 553-570.
20. F. Klopp and J. Ralston, Generic behaviour of open band edges for periodic Schrodinger operators and applications to Lifshits tail asymptotics, work in progress, 1997.
21. P. Kuchment, Floquet Theory for Partial Differential Equations, Oper. Theory Adv. Appl. 60, Birkhauser, Basel, 1993.
22. I. M. Lifshits, Structure of the energy spectrum of impurity bands in disordered solid solutions, Soviet Phys. JETP 17 (1963), 1159-1170.
23. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems, Wiley, New York, 1988.
24. J. N. Mather, “On Nirenberg's proof of Malgrange's preparation theorem” in Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Math. 192, Springer-Verlag, Berlin, 1971.
25. G. Mezincescu, Internal Lifshitz singularities of disordered finite-difference Schrodinger operators, Comm. Math. Phys. 103 (1986), 107-116.
26. G. Mezincescu, Lifshitz singularities for periodic operators plus random potentials, J. Statist. Phys. 49 (1987), 1081-1090.
27. G. Mezincescu, Internal Lifshitz singularities for one-dimensional Schrodinger operators, Comm. Math. Phys. 158 (1993), 315-325.
28. S. Nakao, On the spectral distribution of the Schrodinger operator with random potential, Japan J. Math. (N.S.) 3 (1977), 111-139.
29. A. Outassourt, Comportement semi-classique pour l'opérateur de Schrodinger à potential périodique, J. Funct. Anal. 72 (1987), 65-93.
30. L. Pastur, The behavior of certain Wiener integrals as t ^ and the density of states of Schrodinger equations with random potential (in Russian), Teoret. Mat. Fiz. 32 (1977), 88-95.
31. L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren Math. Wiss. 297, Springer-Verlag, Berlin, 1992.
32. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
33. M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978.
34. M. A. Shubin, Spectral theory and the index of elliptic operators with almost-periodic coefficients (in Russian), Uspekhi Mat. Nauk 34, no. 2 (1979), 95-135;
35. M. A. Shubin, English transl. in Russian Math. Surveys 34, no.2 (1979), 109-157.
36. B. Simon, Schrodinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447-526.
37. B. Simon, Semiclassical analysis of low lying eigenvalues, III: Width of the ground state band in strongly coupled solids, Ann. Physics 158 (1984), 415-420.
38. B. Simon, Lifshitz tails for the Anderson model, J. Statist. Phys. 38 (1985), 65-76.
39. B. Simon, Internal Lifshitz tails, J. Statist. Phys. 46 (1987), 911-918.
40. J. Sjostrand, “Microlocal analysis for the periodic magnetic Schrodinger equation and related questions” in Microlocal Analysis and Applications (Montecatini Terme, 1989), Lecture Notes in Math. 1495, Springer-Verlag, Berlin, 1991, 237-332.
41. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proc. Steklov Inst. Math. 171 (1987), 1-121.