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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket


Authors: Kasso A. Okoudjou and Robert S. Strichartz
Journal: Proc. Amer. Math. Soc. 135 (2007), 2453-2459
MSC (2000): Primary 35P20, 28A80; Secondary 42C99, 81Q10
Published electronically: March 29, 2007
MathSciNet review: 2302566
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket $ SG$. In particular, using the existence of localized eigenfunctions for the Laplacian on $ SG$ we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.


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Additional Information

Kasso A. Okoudjou
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Email: kasso@math.umd.edu

Robert S. Strichartz
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email: str@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09008-9
PII: S 0002-9939(07)09008-9
Keywords: Analysis on fractals, Schr\"odinger operators, Sierpi\'nski gasket
Received by editor(s): January 9, 2006
Published electronically: March 29, 2007
Additional Notes: The research of the second author was supported in part by the National Science Foundation, grant DMS-0140194.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society