Transactions of the American Mathematical Society

Embedded singular continuous spectrum for one-dimensional Schrödinger operators

Author(s): Christian Remling
Journal: Trans. Amer. Math. Soc. 351 (1999), 2479-2497.
MSC (1991): Primary 34L40, 81Q10
Posted: February 24, 1999
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Abstract: We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.

1.: E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. MR 16:1022b
2.: R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. d'Analyse Math. 69 (1996), 153-200. MR 97m:47002
3.: R. del Rio, B. Simon, and G. Stolz, Stability of spectral types for Sturm-Liouville operators, Math. Research Letters 1 (1994), 437-450. MR 95i:47084
4.: D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), 30-56. MR 89a:34033
5.: A. Y. Gordon, S. A. Molchanov, and B. Tsagani, Spectral theory of one-dimensional Schrödinger operators with strongly fluctuating potentials, Funct. Anal. Appl. 25 (1992), 236-238. MR 93a:34097
6.: S. Y. Jitomirskaya and Y. Last, Dimensional Hausdorff properties of singular continuous spectra, Phys. Rev. Letters 76 (1996), 1765-1769. MR 96k:81041
7.: W. Kirsch, S. A. Molchanov, and L. Pastur, One-dimensional Schrödinger operators with high potential barriers. Operator Theory: Advances and Applications, Vol. 57, 163-170. Birkhäuser Verlag, Basel, 1992. MR 94f:34167
8.: A. Kiselev, Y. Last, and B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. Math. Phys. 194 (1998), 1-45. CMP 98:14
9.: Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445. MR 97k:51044
10.: Y. Last and B. Simon, Eigenfunctions, Transfer Matrices, and Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators, to appear in Inv. Math.
11.: S. Molchanov, One-dimensional Schrödinger operators with sparse potentials, preprint (1997).
12.: S. N. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theor. and Math. Phys. 68 (1986), 646-653. MR 88h:81029
13.: D. B. Pearson, Singular Continuous Measures in Scattering Theory, Comm. Math. Phys. 60 (1978), 13-36. MR 58:4076
14.: M. Reed and B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, London-San Diego, 1979. MR 80m:81085
15.: C. Remling, Relationships between the $m$ -function and subordinate solutions of second order differential operators, J. Math. Anal. Appl. 206 (1997), 352-363. MR 97m:34052
16.: C. Remling, A probabilistic approach to one-dimensional Schrödinger operators with sparse potentials, Comm. Math. Phys. 185 (1997), 313-323. MR 98j:3474
17.: C. Remling, Some Schrödinger operators with power-decaying potentials and pure point spectrum, Comm. Math. Phys. 186 (1997), 481-493. MR 98k:34140
18.: C. A. Rogers, Hausdorff Measures, Cambridge University Press, London, 1970. MR 43:7576
19.: W. Rudin, Real and Complex Analysis, McGraw-Hill, Singapore, 1987. MR 88k:00002
20.: B. Simon, Operators with singular continuous spectrum, VII. Examples with borderline time decay, Comm. Math. Phys. 176 (1996), 713-722. MR 97a:47006
21.: B. Simon, Some Schrödinger operators with dense pure point spectrum, Proc. Amer. Math.Soc. 125 (1997), 203-208. MR 97c:34179
22.: B. Simon and G. Stolz, Operators with singular continuous spectrum, V. Sparse potentials, Proc. Amer.Math. Soc. 124 (1996), 2073-2080. MR 97a:47004
23.: G. Stolz, Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal.Appl. 169 (1992), 210-228. MR 93f:34141
24.: W. F. Stout, Almost Sure Convergence, Academic Press, New York, 1974. MR 56:13334
25.: J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Math., Vol. 1258, Springer-Verlag, Berlin-Heidelberg, 1987. MR 89b:47070
26.: A. Zygmund, A remark on Fourier transforms, Proc. Cambridge Phil. Soc. 32 (1936), 321-327.

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34L40, 81Q10

Christian Remling
Affiliation: Universität Osnabrück, Fachbereich Mathematik/Informatik, 49069 Osnabrück, Germany
Email: cremling@mathematik.uni-osnabrueck.de

DOI: 10.1090/S0002-9947-99-02495-2
PII: S 0002-9947(99)02495-2
Keywords: Schr\"odinger equation, singular continuous spectrum, subordinate solutions
Received by editor(s): May 20, 1997
Posted: February 24, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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