Essential spectrum and $L\_2$-solutions of one-dimensional Schrödinger operators
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- by Christian Remling PDF
- Proc. Amer. Math. Soc. 124 (1996), 2097-2100 Request permission
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
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Additional Information
- Christian Remling
- Affiliation: Universität Osnabrück, Fachbereich Mathematik/Informatik, Albrechtstr. 28, D-49069 Osnabrück, Germany
- MR Author ID: 364973
- Email: cremling@chryseis.mathematik.uni-osnabrueck.de
- Received by editor(s): January 23, 1995
- Communicated by: Hal L. Smith
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2097-2100
- MSC (1991): Primary 34L40; Secondary 47E05, 81Q10
- DOI: https://doi.org/10.1090/S0002-9939-96-03463-6
- MathSciNet review: 1342044