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

Abstract

Abstract. In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain iî C R" with fractal boundary 3Q . We construct an open set S for which we can effectively compute the second term of the asymptotics of the "counting function" N(X, S), the number of eigenvalues less than A . In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In X , not to a constant. We also establish some properties of the f-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set t? derived from @ . Analogous periodic functions still appear in our inequalities. These results have been announced in [FV].

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

“…An example is given in [5] consisting of an infinite sequence of square regions in IR 2 , where the boundary has Minkowski dimension D and 1 < D < 2, but the second term of the eigenvalue distribution function oscillates between two constant multiples of A D/2 . However, as is pointed out, the boundary is not Minkowski measurable.…”

“…Such an example can be lifted to dimension 2 by taking the Cartesian product with the open unit interval, just as in [8, example 5-1']. It is claimed in [5] that the infinite sequence of square regions can be altered to produce a counterexample to (1-2) by removing an appropriate sequence of isolated points. If one wishes to remove countable sequences from open sets so as to create counterexamples to (1*2), the procedure we suggest below in Sections 5 and 6 is both simpler, and also serves to disprove the conjecture (1'4).…”

Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

“…However, when Ω is fractal domain, i.e., its boundary Γ is "fractal", the situation will be more complicated (cf. [2,3,5,[9][10][11]). Here in this paper, we shall study the case for Ω is one-dimensional fractal string (see the definition below).…”

Refined asymptotics for eigenvalues on domains of infinite measure

“…This example can be extended to R 2 , by defining Ω = k∈N Ω k , where Ω k consists of m k disjoints squares of sides n 1−k . When Ω has finite measure, similar examples were considered in [5,14,18], where oscillating second term were obtained for the spectral counting function of the Laplace operator in Ω with Dirichlet boundary conditions in the boundary of each square. It is not difficult to extend those arguments to the infinite measure case (that is, m 2 > n), to obtain in this way a quasibounded set with an oscillating main term.…”