Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Imbedded singular continuous spectrum for Schrödinger operators

Author: Alexander Kiselev
Journal: J. Amer. Math. Soc. 18 (2005), 571-603
MSC (2000): Primary 34L40; Secondary 34L25
Published electronically: April 27, 2005
MathSciNet review: 2138138
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct examples of potentials $V(x)$ satisfying $\vert V(x)\vert \leq \frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if $\vert V(x)\vert \leq \frac{B}{1+x},$ the singular continuous spectrum is empty. Therefore our result is sharp.

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

  • 1. R. Carmona, Exponential localization in one-dimensional disordered systems, Duke Math. J. 49 (1982), 191-213. MR 0650377 (84j:82082)
  • 2. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Boston: Birkhäuser, 1990. MR 1102675 (92k:47143)
  • 3. M. Christ and A. Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results, J. Amer. Math. Soc. 11 (1998), 771-797.MR 1621882 (99g:34166)
  • 4. -, WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials, Comm. Math. Phys. 218 (2001), 245-262.MR 1828980 (2002e:34143)
  • 5. -, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal. 12 (2002), 1174-1234.MR 1952927 (2003m:47019)
  • 6. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
  • 7. H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer-Verlag, 1987.MR 0883643 (88g:35003)
  • 8. P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Commun. Math. Phys. 203 (1999), 341-347.MR 1697600 (2000c:34223)
  • 9. S. Denisov, On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potentials, J. Differential Equations 191 (2003), 90-104.MR 1973283 (2004b:34219)
  • 10. F. Germinet, A. Kiselev and S. Tcheremchantsev, Transfer matrices and transport for 1D Schrödinger operators with singular spectrum, Annales de l'Institut Fourier, 54 (2004), 787-830.MR 2097423
  • 11. R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158 (2003), 253-321.MR 1999923 (2004f:47040)
  • 12. A. Kiselev, Y. Last, and B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Commun. Math. Phys. 194 (1998), 1-45. MR 1628290 (99g:34167)
  • 13. T. Kriecherbauer and C. Remling, Finite gap potentials and WKB asymptotics for one-dimensional Schrödinger operators, Comm. Math. Phys. 223 (2001), 409-435.MR 1864439 (2003a:81046)
  • 14. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445. MR 1423040 (97k:81044)
  • 15. B. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht 1987.MR 0933088 (89b:34001)
  • 16. V. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986.MR 0897106 (88f:34034)
  • 17. S.N. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theor.-math. 68 (1986), 18-28.MR 0875178 (88h:81029)
  • 18. D. Pearson, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), 13-36.MR 0484145 (58:4076)
  • 19. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I. Functional Analysis, Academic Press, New York, 1972.MR 0493419 (58:12429a)
  • 20. -, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, London-San Diego, 1979.MR 0529429 (80m:81085)
  • 21. C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151-170.MR 1620313 (99f:34123)
  • 22. -, Bounds on embedded singular spectrum for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 128 (2000), no. 1, 161-171.MR 1637420 (2000c:34225)
  • 23. -, Schrödinger operators with decaying potentials: some counterexamples, Duke Math. J. 105 (2000), no. 3, 463-496.MR 1801769 (2002e:81033)
  • 24. B. Simon, Schrödinger operators in the twenty-first century, Mathematical Physics 2000, 283-288, Imp. Coll. Press, London, 2000. MR 1773049 (2001g:81071)
  • 25. -, Some Schrödinger operators with dense point spectrum, Proc. Amer. Math. Soc. 125 (1997), 203-208. MR 1346989 (97c:34179)
  • 26. -, Operators with singular continuous spectrum. I. General operators, Ann. of Math. (2) 141 (1995), no. 1, 131-145. MR 1314033 (96a:47038)
  • 27. J. von Neumann and E.P. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465-467.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 34L40, 34L25

Retrieve articles in all journals with MSC (2000): 34L40, 34L25

Additional Information

Alexander Kiselev
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388

Keywords: Schr\"odinger operators, scattering, singular spectrum
Received by editor(s): November 14, 2003
Published electronically: April 27, 2005
Additional Notes: The author was supported in part by NSF grant DMS-0314129 and by an Alfred P. Sloan Research Fellowship
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society