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Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

Singular continuous spectrum is generic

Author(s): R. Del Rio; S. Jitomirskaya; N. Makarov; B. Simon
Journal: Bull. Amer. Math. Soc. 31 (1994), 208-212.
MathSciNet review: 1260519
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Abstract | References | Additional information

Abstract: In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense $ {G_\delta }$.

References:

Bibliography

[1]
S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 151-218. MR 0397194 (53:1053)

[2]
M. Aizenman and S. Molchanov, Localization at large disorders and at extreme energies: An elementary derivation, Comm. Math. Phys. 157 (1993), 245-278. MR 1244867 (95a:82052)

[3]
E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilation-analytic interactions, Comm. Math. Phys. 22 (1971), 280-294. MR 0345552 (49:10288)

[4]
R. Carmona, Exponential localization in one-dimensional disordered systems, Duke Math. J. 49 (1982), 191-213. MR 650377 (84j:82082)

[5]
R. del Rio Castillo, A forbidden set for embedded eigenvalues, Proc. Amer. Math. Soc. 121 (1994), 77-82. MR 1191867 (94g:34147)

[6]
R. del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum, II. Rank one operators, Comm. Math. Phys. (to appear). MR 1298942 (97a:47002)

[7]
F. Delyon, H. Kunz, and B. Souillard, One-dimensional wave equations in disordered media, J. Phys. A 16 (1983), 25-42. MR 700179 (84g:81022)

[8]
F. Delyon, Y. Lévy, and B. Souillard, Anderson localization for multi-dimensional systems at large disorder or large energy, Comm. Math. Phys. 100 (1985), 463-470. MR 806247 (86m:82042)

[9]
J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151-189. MR 696803 (85c:82004)

[10]
J. Fröhlich, T. Spencer, and P. Wittwer, Localization for a class of one dimensional quasiperiodic Schrödinger operators, Comm. Math. Phys. 132 (1990), 5-25. MR 1069198 (91h:35095)

[11]
I. Goldsheid, Asymptotics of the product of random matrices depending on a parameter, Soviet Math. Dokl. 16 (1975), 1375-1379.

[12]
I. Goldsheid, S. Molchanov, and L. Pastur, A pure point spectrum of the stochastic one-dimensional Schrödinger equation, Funct. Anal. Appl. 11 (1977), 1-10. MR 0470515 (57:10266)

[13]
A. Gordon, On exceptional value of the boundary phase for the Schrödinger equation of a half-line, Russian Math. Surveys 47 (1992), 260-261.

[14]
-, Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Comm. Math. Phys. (to appear). MR 1291242 (95k:47019)

[15]
P. Halmos, In general a measure preserving transformation is mixing, Ann. Math. 45 (1944), 786-792. MR 0011173 (6:131d)

[16]
S. Jitomirskaya, Anderson localization for the almost Mathieu equation: A nonperturbative proof, Comm. Math. Phys. (to appear). MR 1298941 (95i:81045)

[17]
T. Kato, Perturbation theory for linear operators (2nd, ed.), Springer-Verlag, Berlin, Heidelberg, and New York, 1980.

[18]
S. Kotani and B. Simon, Localization in general one-dimensional random systems, II. Continuum Schrödinger operators, Comm. Math. Phys. 112 (1987), 103-119. MR 904140 (89d:81034)

[19]
P. Perry, I. Sigal, and B. Simon, Spectral analysis of N-body Schrödinger operators, Ann. of Math. (2) 114 (1981), 519-567. MR 634428 (83b:81129)

[20]
M. Reed and B. Simon, Methods of modern mathematical physics, IV. Analysis of operators, Academic Press, London, 1978. MR 0493421 (58:12429c)

[21]
V. Rohlin, A "general" measure-preserving transformation is not mixing, Dokl. Akad. Nauk SSSR (N.S.) 60 (1948), 349-351. MR 0024503 (9:504d)

[22]
B. Simon, Operators with singular continuous spectrum, I. General operators, Ann. of Math. (to appear). MR 1314033 (96a:47038)

[23]
B. Simon and S. Jitomirskaya, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Comm. Math. Phys. (to appear). MR 1298948 (97a:47003)

[24]
B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75-90. MR 820340 (87k:47032)

[25]
Ya. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Stat. Phys. 46 (1987), 861-909. MR 893122 (88h:82016)

[26]
J. von Neumann, Charakterisierung des Spektrums eines Integraloperators, Actualités Sci. Indust., no. 229, Hermann, Paris, 1935.

[27]
H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo 27 (1909), 373-392.

Additional Information:

DOI: 10.1090/S0273-0979-1994-00518-X
PII: S 0273-0979(1994)00518-X
Copyright of article: Copyright 1994, American Mathematical Society




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