Lifshitz tails for random Schrödinger operators with negative singular Poisson potential

Communications in Mathematical Physics

Volumn 206, Issue 1, 1999, Pages 57-103

  Klopp, Frédéric ^a   Pastur, Leonid ^b,c  

^a UNIVERSITÉ PARIS 13   (France)

^b UNIVERSITÉ PARIS DESCARTES   (France)

^c INSTITUTE OF HYDROBIOLOGY   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033450317     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050698     Document Type: Article

References (32)

1. 1. Combes, J.M. and Hislop, P.D.: Localization for some continuous random hamiltonians in d-dimensions. J. Funct. Anal. 124, 149-180 (1994)
2. 2. Cycon, H.L., Froese, R.G., Kirsch, W. and Simon, B.: Schrödinger Operators. Berlin: Springer Verlag, 1987
3. 3. Donsker, M. and Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure and App. Math. 28, 525-565 (1975)
4. 4. Efros, M. and Shlovski, B.: Electronic properties of doped semi-conductors. Heidelberg: Springer Verlag, 1984
5. 5. Helffer, B. and Sjöstrand, J.: On diamagnetism and the De Haas-Van Alphen effect. Ann. de l'Institut Henri Poincaré, série Phys. Théor. 52, 303-375 (1990)
6. 6. Hörmander, L.: The analysis of linear partial differential equations. I. Vol. 256 of Grundlehren der Mathematischen Wissenschaften, Berlin-Heidelberg-New York: Springer Verlag, 1990
7. 7. Kirsch, W.: Random Schrödinger operators. In: A. Jensen H. Holden, editor, Schrödinger Operators, Number 345 in Lecture Notes in Physics, Berlin: Springer Verlag, 1989
8. 8. Klopp, F.: An asymptotic expansion for the density of states of a random Schrödinger operator with Bernoulli disorder. Random Operators and Stochastic Equations 3 (4), 315-332 (1995)
9. 9. Klopp, F.: A low concentration asymptotic expansion for the density of states of a random Schrödinger operator with Poisson disorder. J. Funct. Anal. 145, 267-295 (1995)
10. 10. Klopp, F.: Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1. J. Stat. Phy. 90 (3-4), 927-947 (1998)
11. 11. Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J. 1999. To appear
12. 12. Klopp, F.: Precise high energy asymptotics for the integrated density of states of an unbounded random Jacobi matrix. Rev. Math. Phys. 1999. To appear
13. 13. Klopp, F. and Pastur, L.: In progress
14. 14. Kuchment, P.: Floquet theory for partial differential equations. Vol. 60 of Operator Theory: Advances and Applications, Basel: Birkhäuser, 1993
15. 15. Landau, L. and Lifshitz, L. Mécanique quantique, théorie non-relativiste. Moscou: Editions MIR, 1966
16. 16. Lifshitz, I.M.: Structure of the energy spectrum of impurity bands in disordered solid solutions. Sov. Phys. JETP 17, 1159-1170 (1963)
17. 17. Lifshitz, I.M.: Energy spectrum structure and quantum states of disordered condensed systems. Sov. Phys. Uspekhi 7, 549-573 (1965)
18. 18. Lifshitz, I.M., Gredeskul, S.A. and Pastur, L.A. Introduction to the theory of disordered systems. New-York: Wiley, 1988
19. 19. Mather, J.N. On Nirenberg's proof of Malgrange's preparation theorem. In: Proceedings of Liverpool Singularities-Symposium I, Number 192 in Lecture Notes in Mathematics, Berlin: Springer Verlag, 1971
20. 20. Pastur, L.: Behaviour of some Wiener integrals as t → +∞ and the density of states of the Schrödinger equation with a random potential. Teor.-Mat.-Fiz 32, 88-95 (1977) (in Russian)
21. 21. Pastur, L. and Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin: Springer Verlag, 1992
22. 22. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators. New-York: Academic Press, 1978
23. 23. Reed, M. and Simon, B. Methods of Modern Mathematical Physics, Vol I: Functional Analysis. New-York: Academic Press, 1980
24. 24. Shubin, M.A. Spectral theory and index of elliptic operators with almost periodic coefficients. Russ. Math. Surv. 34, 109-157 (1979)
25. 25. Simon, B.: Trace ideals and their applications. Cambridge: Cambridge University Press, 1979
26. 26. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447-526 (1982)
27. 27. Simon, B.: Lifshitz tails for the Anderson model. J. Stat. Phys. 38, 65-76 (1985)
28. 28. Sjöstrand, J.: Microlocal analysis for periodic magnetic Schrödinger equation and related questions. In: Microlocal analysis and applications, Vol. 1495 of Lecture Notes in Mathematics Berlin: Springer Verlag, 1991
29. 29. Stein, E.: Singular integrals and Differentiability properties of functions. Princeton, N.J.: Princeton University Press, 1970
30. 30. Sznitman, A.: Lifshitz tails and Wiener sausages. I. J. Funct. Anal. 94, 223-246 (1990)
31. 31. Sznitman, A.: Fluctuations of principal eigenvalues and random scales. Commun. Math. Phys. 189, 337-363 (1997)
32. 32. Taylor, M. Partial differential equations. New-York-Berlin: Springer, 1996