AN EXAMPLE OF A 2-TERM ASYMPTOTICS FOR THE COUNTING FUNCTION OF A FRACTAL DRUM

In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain OMEGA subset-of R(n) with fractal boundary partial derivative OMEGA. We construct an open set Q for which we can effectively compute the second term of the asymptotics of the ''counting function'' N(lambda, 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 lnlambda, 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 O derived from Q. Analogous periodic functions still appear in our inequalities. These results have been announced in [FV].