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

 

Compactness of isospectral potentials


Author: Harold Donnelly
Journal: Trans. Amer. Math. Soc. 357 (2005), 1717-1730
MSC (2000): Primary 58G25
Published electronically: December 29, 2004
MathSciNet review: 2115073
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Abstract: The Schrödinger operator $-\Delta+V$, of a compact Riemannian manifold $M$, has pure point spectrum. Suppose that $V_0$ is a smooth reference potential. Various criteria are given which guarantee the compactness of all $V$satisfying $\operatorname{spec}(-\Delta+V)=\operatorname{spec}(-\Delta+V_0)$. In particular, compactness is proved assuming an a priori bound on the $W_{s,2}(M)$ norm of $V$, where $s>n/2-2$ and $n=\dim M$. This improves earlier work of Brüning. An example involving singular potentials suggests that the condition $s>n/2-2$ is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions $n\le 9$.


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Additional Information

Harold Donnelly
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03813-9
PII: S 0002-9947(04)03813-9
Received by editor(s): April 28, 2003
Published electronically: December 29, 2004
Additional Notes: The author was partially supported by NSF Grant DMS-0203070
Article copyright: © Copyright 2004 American Mathematical Society