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 bo unded domain OMEGA subset-of R(n) with fractal boundary partial deriva tive OMEGA. We construct an open set Q for which we can effectively co mpute the second term of the asymptotics of the ''counting function'' N(lambda, Q), the number of eigenvalues less than lambda. In this exam ple, contrary to the M. V. Berry conjecture, the second asymptotic ter m is proportional to a periodic function of lnlambda, not to a constan t. We also establish some properties of the zeta-function of this prob lem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set O derived from Q. Analogous perio dic functions still appear in our inequalities. These results have bee n announced in [FV].