Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators

Author(s): Dirk Buschmann; Günter Stolz
Journal: Trans. Amer. Math. Soc. 353 (2001), 635-653.
MSC (2000): Primary 81Q10, 34L40, 60H25, 47B80
Posted: October 19, 2000
MathSciNet review: 1804511
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.

References:

1.: G. Borg: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Math. 78, 1 - 96 (1946) MR 7:382d
2.: D. Buschmann: A proof of the Ishii-Pastur theorem by the method of subordinacy. Univ. Iagel. Acta Math. 34, 29 - 34 (1997) MR 98i:82031
3.: R. Carmona: Exponential localization in one-dimensional disordered systems. Duke Math. J. 49, 191 - 213 (1982) MR 84j:82082
4.: R. Carmona, A. Klein and F. Martinelli: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys., 108, 41 - 66 (1987) MR 88f:82087
5.: R. Carmona and J. Lacroix: Spectral theory of random Schrödinger operators. Birkhäuser, Basel-Berlin 1990 MR 94k:47143
6.: E.A. Coddington and N. Levinson: Theory of ordinary differential equations. McGraw-Hill, New York 1955 MR 16:1022b
7.: J.-M. Combes and P.D. Hislop: Localization for some continuous random Hamiltonians in -dimensions. J. Funct. Anal. 124, 149 - 180 (1994) MR 95g:82047
8.: J.M. Combes, P.D. Hislop and E. Mourre: Spectral averaging, perturbation of singular spectrum, and localization. Trans. Amer. Math. Soc. 348, 4883 - 4894 (1996) MR 97c:35145
9.: H. von Dreifus and A. Klein: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285 - 299 (1989) MR 90k:82056
10.: D.J. Gilbert: On subordinacy and analysis of Schrödinger operators with two singular endpoints. Proc. Royal Soc. Edinburgh 112A, 213 - 229 (1989) MR 90j:47061
11.: I.Ya. Goldsheidt, S. Molchanov and L. Pastur: A pure-point spectrum of the stochastic one-dimensional Schrödinger equation. Funct. Anal. Appl. 11, 1 - 8 (1977) MR 57:10266
12.: A. Figotin and A. Klein: Localization of classical waves I: Acoustic waves. Commun. Math. Phys. 180, 439 - 482 (1996) MR 2000d:35240a
13.: B.V. Gnedenko: The theory of probability, 4th Edition, Chelsea Publishing Company, New York, 1967 MR 36:913
14.: W. Kirsch, S. Kotani and B. Simon: Absence of absolutely continuous spectrum for some one dimensional random but deterministic potentials. Ann. Inst. Henri Poincaré Phys. Théor. 42, 383 - 406 (1985) MR 87h:60115
15.: W. Kirsch, P. Stollmann and G. Stolz: Anderson localization for random Schrödinger operators with long range interactions. Commun. Math. Phys. 195, 495-507 (1998) MR 99e:82051
16.: F. Klopp: Localization for semiclassical continuous random Schrödinger Operators II: The random displacement model. Helv. Phys. Acta 66, 810 - 841 (1993) MR 96c:35204
17.: F. Klopp: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167, 553 - 569 (1995) MR 95m:82080
18.: S. Kotani: Lyapunov indices determine absolutely continuous spectra of stationary one dimensional Schrödinger operators. In: Stochastic Analysis, ed. K. Ito, 225 - 247, North Holland, Amsterdam 1984 MR 86h:60117
19.: S. Kotani: Lyapunov exponents and spectra for one-dimensional random Schrödinger operators. In: Random Matrices and their Applications, eds. J.E. Cohen, H. Kesten and C.M. Newman, American Mathematical Society, Contemporary Math. 50, 277 - 286, Providence 1986 MR 88a:60116
20.: S. Kotani: One-dimensional random Schrödinger operators and Herglotz functions. In: Probabilistic Methods in Mathematical Physics, eds. K. Ito and N. Ikeda, 219 - 250, Academic Press, Boston 1987 MR 89h:60100
21.: S. Kotani and B. Simon: Localization in general one-dimensional random systems. Commun. Math. Phys. 112, 103 - 119 (1987) MR 89d:81034
22.: H. Kunz and B. Souillard: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 79, 201 - 249 (1980) MR 83f:39003
23.: N. Minami: An extension of Kotani's theorem to random generalized Sturm-Liouville operators. Commun. Math. Phys. 103, 387 - 402 (1986) MR 87j:81042
24.: N. Minami: An extension of Kotani's theorem to random generalized Sturm-Liouville operators II. In: Stochastic processes in classical and quantum systems, 411 - 419, Lecture Notes in Physics 262, Springer 1986 MR 88h:81028
25.: N. Minami: Random Schrödinger operators with a constant electric field. Ann. Inst. Henri Poincaré Phys. Théor. 56, 307 - 344 (1992) MR 93d:34142
26.: S. Molchanov: Lectures on random media. In: Lectures on probability theory (Saint-Flour, 1992), 242-411, Lecture Notes in Math. 1581, Springer 1994. MR 95m:60165
27.: L. Pastur and A. Figotin: Spectra of random and almost-periodic operators. Springer-Verlag, 1991 MR 94h:47068
28.: M. Reed and B. Simon: Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, New York 1978 MR 58:12429c
29.: B. Simon: Localization in general one dimensional random systems, I. Jacobi matrices. Commun. Math. Phys. 102, 327 - 336 (1985) MR 87h:81039
30.: B. Simon and T. Wolff: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math. 39, 75 - 90 (1986) MR 87k:47032
31.: G. Stolz: Spectral theory of Schrödinger operators with potentials of infinite barriers type. Habilitationsschrift, Frankfurt 1994
32.: G. Stolz: Localization for random Schrödinger operators with Poisson potential. Ann. Inst. Henri Poincaré Phys. Théor. 63, 297 - 314 (1995) MR 96h:34174

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|