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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniqueness results for one-dimensional Schrödinger operators with purely discrete spectra


Authors: Jonathan Eckhardt and Gerald Teschl
Journal: Trans. Amer. Math. Soc. 365 (2013), 3923-3942
MSC (2010): Primary 34B20, 34L05; Secondary 34B24, 47A10
Published electronically: November 27, 2012
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Abstract: We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt-Lieberman type result for these operators. Our approach is based on the singular Weyl-Titchmarsh-Kodaira theory which is extended to cover the present situation.


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Additional Information

Jonathan Eckhardt
Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria
Email: jonathan.eckhardt@univie.ac.at

Gerald Teschl
Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Wien, Austria — and — International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
Email: Gerald.Teschl@univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05821-1
PII: S 0002-9947(2012)05821-1
Keywords: Schr\"odinger operators, inverse spectral theory, discrete spectra
Received by editor(s): October 11, 2011
Received by editor(s) in revised form: February 28, 2012
Published electronically: November 27, 2012
Additional Notes: This research was supported by the Austrian Science Fund (FWF) under Grant No. Y330
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.