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THE BEHAVIOUR OF THE SPECTRAL COUNTING FUNCTION FOR A FAMILY OF SETS WITH FRACTAL BOUNDARIES

Published online by Cambridge University Press:  01 February 1997

JAMES FAWKES
JAMES FAWKES
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail: [email protected]
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Abstract

We construct a one-parameter family of sets in ℝ2 generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Minkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 126 - 138
Copyright
The London Mathematical Society 1997

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