Compactness of isospectral potentials
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- by Harold Donnelly PDF
- Trans. Amer. Math. Soc. 357 (2005), 1717-1730 Request permission
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 $n\le 9$.References
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Additional Information
- Harold Donnelly
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Received by editor(s): April 28, 2003
- Published electronically: December 29, 2004
- Additional Notes: The author was partially supported by NSF Grant DMS-0203070
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1717-1730
- MSC (2000): Primary 58G25
- DOI: https://doi.org/10.1090/S0002-9947-04-03813-9
- MathSciNet review: 2115073