Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps

We study the eigenvalue asymptotics of a Neumann Laplacian −Δ_N^Ω in unbounded regions Ω of R^2 with cusps at infinity (a typical example is Ω = {(x, y) ϵR^2: x > 1, ¦y¦< e^(−x)^2}) and prove that N_E(−Δ_N^Ω) ~ N_E(H_v) +E2Vol(Ω), where H_v is the canonical one-dimensional Schrödinger operator associated to the problem. We establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a Dirichlet Laplacian −Δ_D^Ω for a class of cusp-type regions of infinite volume

Spectral Properties of Random Schrödinger Operators with Unbounded Potentials

We investigate spectral properties of random Schrödinger operators H_ω = - Δ + ξ_n(ω)(1 + │n│^ɑ) acting on l^2(Z^d), where ξ_n, are independent random variables uniformly distributed on [0, 1]

Spectral Properties of Random Schrödinger Operators with Unbounded Potentials

We investigate spectral properties of random Schrödinger operators H_ω = - Δ + ξ_n(ω)(1 + │n│^ɑ) acting on l^2(Z^d), where ξ_n, are independent random variables uniformly distributed on [0, 1]