An example of a two-term asymptotics for the “counting function” of a fractal drum

Abstract:

In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $\Omega \subset {\mathbb {R}^n}$ with fractal boundary $\partial \Omega$. We construct an open set $\mathcal {Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $N(\lambda ,\mathcal {Q})$, the number of eigenvalues less than $\lambda$. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $\lambda$, not to a constant. We also establish some properties of the $\zeta$-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $\mathcal {O}$ derived from $\mathcal {Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $[{\text {FV}}]$.