We explicitely compute the absolutely continuous spectrum of the Laplace-Beltrami operator for p-forms for the class of warped product metrics dσ = ydy + ydθ S , where y is a boundary defining function on the unit ball B(0, 1) in R .

We study the dispersive properties of the wave equation associated with the shifted Laplace–Beltrami operator on real hyperbolic spaces and deduce new Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well–posedness results for the nonlinear wave equation.

We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods, we study the spectrum of the Laplace–Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of Γ0(5)-cusp forms is presented.

The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the two-point spectral correlation functions of Riemann zeta function zeros, and of the Laplace–Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos applications and can serve as a non-formal introduction to mathematical methods of quantum chaos.

We consider \geometric" scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between two half-lines. We discuss applicability and correctness of this model. With an example, we show that such scattering problem may exhibit unusual properties: the transition coeecient has a sequence of sharp peaks which become more and more distant at high energy and otherwise tur...

For any given momentum vector $\xi _0\in \mathbb {S}^1$ we construct a continuous family of quasimodes for the Laplace–Beltrami operator on translation surface. We apply our result to rational polygonal quantum billiards and thus

We consider numerical approximations of spectral fractional Laplace-Beltrami problems on closed surfaces. The proposed algorithms rely their Balakrishnan integral representation and consist a sinc quadrature coupled with standard finite element methods for parametric Possibly up to log term, optimal rates convergence are observed derived analytically when the discrepancies between exact solutio...

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed f...