Asymptotic Trace Of Schr(?)dinger Operator On Fractal Drums

In 1953, L.M.Gelfand and B.M.Levitan studied the trace of the Sturm — Liouville problemwhere q(x) is a bounded and differentiable function on [0, π]. They proved the eigenvalues satisfy the following identity:Starting from this point, many progesses has mad on the study to Sturm— Liouville operator. In 1979, Cao Cewen extended the problem to higher dimensional cases, that is the case of trace problem of Schrodinger operator:where Ω is a bounded and connected domain with piecewise smooth boundary Ω, q(x) is a bounded and differentiable function in SI. Let λ_j, be the j-th eigenvalue, μ_j be the j-th eigenvalue of the Dirichlet — Laplacian when g(x) = 0, then there exists R→∞ such thatCao Cewen aslo extended the Gelf and — Levitan's identity, and proved when ft is an n—dimensional cube:where In the present paper, we extend Cao's resuts, and consider the case of a non-connected domain ft with fractal boundaries, i.e., is an open subset of R~n with boundary Ω, and the connected regions Ω_m with piecewise smooth boundaries are bounded and pairwise disjoint. We assume that denotes the n-dimensional Lesbegues measure . q(x) is continuous function on Ω_m, but it does not mean q(x) is also continuous on ft. We have the following main results: (l)there exists R →∞, such that(2)when n = 1, the following trace displacement holdswhere the boundary 9ft is Minkowski measureable and has Minkowski dimension D ∈ (0,1), denote then Riernann - zeta function, M{D, Ω) is the Minkowski content of theboundary( 3 ) whenn = 2, we firstly get the following average trace displacement formula:where the Weylterm. Finally, for some special case, we get also the trace displacement estimates which is similar to the case (2), that means...