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Bounds on embedded singular spectrum
for one-dimensional Schrödinger operators


Author: Christian Remling
Journal: Proc. Amer. Math. Soc. 128 (2000), 161-171
MSC (1991): Primary 34L40, 81Q10
DOI: https://doi.org/10.1090/S0002-9939-99-05110-2
Published electronically: June 24, 1999
MathSciNet review: 1637420
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Abstract: We show that the solutions of the one-dimensional Schrödinger equation $-y''+Vy=Ey$ with potential $V(x)=O(x^{-\alpha})$ satisfy the WKB asymptotic formulae off a set of energies $E$ of Hausdorff dimension $\le 2(1-\alpha)$. This result gives restrictions on the structure of possible embedded singular spectrum. The proof relies on new norm estimates for the integral transform associated with the WKB method.


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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-99-05110-2
Received by editor(s): March 10, 1998
Published electronically: June 24, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society

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