相关论文: An Algorithm for the Computation of Eigenvalues, S…
The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions and holomorphic $s$-differentials satisfy certain consistency conditions on closed hyperbolic surfaces. These consistency conditions can be…
The Laplace operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions…
We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maass-Laplace operators is…
This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction $u$ as a linear combination of eigenfunctions corresponding to the common…
These are lecture notes from a series of three lectures given at the summer school "Geometric and Computational Spectral Theory" in Montreal in June 2015. The aim of the lecture was to explain the mathematical theory behind computations of…
In this paper we analyze the spectral zeta function associated with a Laplace operator acting on scalar functions on an N-dimensional Euclidean space in the presence of a spherically symmetric background potential. The obtained analytic…
We obtain a sharp lower estimate on eigenvalues of Laplace--Beltrami operator on a hyperbolic surface with injectivity radius bounded from the below.
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…
In this paper, we propose a meshless method of computing eigenvalues and eigenfunctions of a given surface embedded in $\mathbb R^3$. We use point cloud data as input and generate the lattice approximation for some neighborhood of the…
In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have…
The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this…
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based…
Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the…
We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation…
The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space.…
In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type $I\times_{f} N$ where $I$ is an interval of the real line and $N$ is a…
The functional determinant of Laplace-type operators on the 3-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is computed by quadratures and making use of the related terms of the Selberg trace…
We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace-Beltrami operator on functions and the Laplacian on powers of the cotangent bundle.…
For a Riemann surface with cusps we define a theta function using the eigenvalues of the Laplacian and the singularities of the scattering determinant. We provide its meromorphic continuation and discuss its singularities.
We prove an analogue of Selberg's trace formula for a delta potential on a hyperbolic surface of finite volume. For simplicity we restrict ourselves to surfaces with at most one cusp, but our methods can easily be extended to any number of…