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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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: http://dx.doi.org/10.1090/S0002-9947-96-01579-6
PII: S 0002-9947(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