Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embedded singular continuous spectrum for one-dimensional Schrödinger operators


Author: Christian Remling
Journal: Trans. Amer. Math. Soc. 351 (1999), 2479-2497
MSC (1991): Primary 34L40, 81Q10
DOI: https://doi.org/10.1090/S0002-9947-99-02495-2
Published electronically: February 24, 1999
MathSciNet review: 1665336
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 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.

Similar Articles

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

Retrieve articles in all journals with MSC (1991): 34L40, 81Q10


Additional Information

Christian Remling
Email: cremling@mathematik.uni-osnabrueck.de

DOI: https://doi.org/10.1090/S0002-9947-99-02495-2
Keywords: Schr\"odinger equation, singular continuous spectrum, subordinate solutions
Received by editor(s): May 20, 1997
Published electronically: February 24, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society