Spectral averaging, perturbation of singular spectra, and localization

Combes, J.; Hislop, P.; Mourre, E.

doi:10.1090/S0002-9947-96-01579-6

Spectral averaging, perturbation of singular spectra, and localization
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by J. M. Combes, P. D. Hislop and E. Mourre PDF

Trans. Amer. Math. Soc. 348 (1996), 4883-4894 Request permission

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.

References

H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643, DOI 10.1007/978-3-540-77522-5
J.-M. Combes and P. D. Hislop, Localization for some continuous, random Hamiltonians in $d$-dimensions, J. Funct. Anal. 124 (1994), no. 1, 149–180. MR 1284608, DOI 10.1006/jfan.1994.1103
J. M. Combes, P. D. Hislop, C. A. Shubin, and A. Tip, in preparation.
René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
James S. Howland, Perturbation theory of dense point spectra, J. Funct. Anal. 74 (1987), no. 1, 52–80. MR 901230, DOI 10.1016/0022-1236(87)90038-3
James S. Howland, Quantum stability, Schrödinger operators (Aarhus, 1991) Lecture Notes in Phys., vol. 403, Springer, Berlin, 1992, pp. 100–122. MR 1181243, DOI 10.1007/3-540-55490-4_{7}
James S. Howland, On a theorem of Carey and Pincus, J. Math. Anal. Appl. 145 (1990), no. 2, 562–565. MR 1038178, DOI 10.1016/0022-247X(90)90421-B
Frédéric Klopp, Localization for some continuous random Schrödinger operators, Comm. Math. Phys. 167 (1995), no. 3, 553–569 (English, with English and French summaries). MR 1316760, DOI 10.1007/BF02101535
S. Kotani and B. Simon, Localization in general one-dimensional random systems. II. Continuum Schrödinger operators, Comm. Math. Phys. 112 (1987), no. 1, 103–119. MR 904140, DOI 10.1007/BF01217682
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
R. Del Rio, S. Jitomirskaya, N. Makarov, and B. Simon, Singular continuous spectrum is generic, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 208–212. MR 1260519, DOI 10.1090/S0273-0979-1994-00518-X
Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105