29 research outputs found
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
3D oxalate-based coordination polymers: Relationship between structure, magnetism and color, studied by high-field ESR spectroscopy
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]