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| ON A PAIR OF NON-ISOMETRIC ISOSPECTRAL DOMAINS WITHFRACTAL BOUNDARIES AND THE WEYL-BERRY CONJECTURE |
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Citation： |
Sleeman, B. D,Chen Hua.ON A PAIR OF NON-ISOMETRIC ISOSPECTRAL DOMAINS WITHFRACTAL BOUNDARIES AND THE WEYL-BERRY CONJECTURE[J].Chinese Annals of Mathematics B,1998,19(1):9~20 |
| Page view： 1079
Net amount： 791 |
Authors： |
Sleeman, B. D; Chen Hua |
Foundation： |
Project supported by the National Natural Science
Foundation of China and the Royal Society of London |
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| Abstract： |
This paper is divided into two parts. In the first part the
authors extend Kac's
classical problem to the fractal case, i.e., to ask: Must two
isospectral planar domains with fractal boundaries be isometric?
It is demonstrated that the answer to this question is no, by constructing a
pair of disjoint isospectral planar domains whose boundaries have the
same interior Bouligand-Minkowski dimension but are not isometric. In
the second part of this paper the authors give the exact two-term asymptotics
for the Dirichlet counting functions associated with the examples
given here and obtain sharp two sided estimates for the second term of
the counting functions. The first result in the second part of the
paper shows that the coefficient of the second term is an oscillatory
function of $\l$, which implies that the Weyl-Berry
conjecture, for the examples given here, is false. The second result
implies that the weaker form of the Weyl-Berry conjecture, for these
examples, is true. This in turn means that the interior
Bouligand-Minkowski dimension of the examples is a spectral invariant. |
Keywords： |
Non-isometric, Isospectral domain, Fractal boundary,
Weyl-Berry conjecture |
Classification： |
35J25, 35P15, 35P20 |
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