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Essential spectrum and $L_2$-solutions of one-dimensional Schrödinger operators

Author: Christian Remling
Journal: Proc. Amer. Math. Soc. 124 (1996), 2097-2100
MSC (1991): Primary 34L40; Secondary 47E05, 81Q10
MathSciNet review: 1342044
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Abstract: In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of $L_2$-solutions. The proof of this theorem is based on inverse spectral theory.

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

  • 1. Joachim Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. MR 923320
  • 2. P. Hartman and A. Wintner, A seperation theorem for continuous spectra, Amer. J. Math. 71 (1949), 650--662. MR 11:109i
  • 3. B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
  • 4. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 16:1022b

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Additional Information

Christian Remling
Affiliation: Universität Osnabrück, Fachbereich Mathematik/Informatik, Albrechtstr. 28, D-49069 Osnabrück, Germany

Keywords: One-dimensional Schr\"odinger operator, Hartman-Wintner conjecture, $L_2$-solution, essential spectrum, inverse spectral theory
Received by editor(s): January 23, 1995
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society