Spectral averaging, perturbation

of singular spectra, and localization

Authors:
J. M. Combes, P. D. Hislop and E. Mourre

Journal:
Trans. Amer. Math. Soc. **348** (1996), 4883-4894

MSC (1991):
Primary 35P20, 81Q10

DOI:
https://doi.org/10.1090/S0002-9947-96-01579-6

MathSciNet review:
1344205

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Abstract | References | Similar Articles | Additional Information

Abstract: A spectral averaging theorem is proved for one-parameter families of self-adjoint operators using the method of differential inequalities. This theorem is used to establish the absolute continuity of the averaged spectral measure with respect to Lebesgue measure. This is an important step in controlling the singular continuous spectrum of the family for almost all values of the parameter. The main application is to the problem of localization for certain families of random Schrödinger operators. Localization for a family of random Schrödinger operators is established employing these results and a multi-scale analysis.

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Additional Information

**J. M. Combes**

Affiliation:
Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria;
Permanent address (J.M.C.): Départment de Mathématiques, Université de Toulon et du Var, 83130 La Garde, France

**P. D. Hislop**

Affiliation:
Permanent address (P.D.H.): Mathematics Department, University of Kentucky, Lexington, Kentucky 40506-0027

**E. Mourre**

Affiliation:
Centre de Physique Théorique, CNRS, Luminy, France

DOI:
https://doi.org/10.1090/S0002-9947-96-01579-6

Received by editor(s):
August 3, 1994

Received by editor(s) in revised form:
March 20, 1995

Additional Notes:
The second author was supported in part by NSF grants INT 90-15895 and DMS 93-07438

Article copyright:
© Copyright 1996
American Mathematical Society