Discreteness of the spectrum of the LaplaceBeltrami operator Mark Harmer Department of Mathematics Australian National University Australia email: Abstract We propose simple conditions equivalent to the discreteness of the spectrum of the LaplaceBeltrami operator on a class of Riemannian manifolds close to warped products. Since the spectrum of the LaplaceBeltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of nonrigid shapes, i.
e. bendable objects such as humans, animals, plants, etc. Convergence, Stability, and Discrete Approximation of Laplace Spectra Tamal K. Deyy Pawas Ranjany Yusu methods is the LaplaceBeltrami operator of a manifold.
Indeed, a variety of work in graphics and geometric optimization uses the eigenstructures (i. e, the eigenvalues and eigen derstand the stability of the spectrum of the Laplace taking the eigenvalues (i. e. the spectrum) of its LaplaceBeltrami operator.
Employing the LaplaceBeltrami spectra (not the spectra of the mesh Laplacian) as ngerprints of surfaces and solids is a novel approach.
Jun 03, 2018 The first notion of a Laplace operator for functionals on a Hilbert space was introduced by and a lower bound on the spectrum of Laplace operators (Theorem 3. 10). 2008 LaplaceBeltrami operator, Laplacede Rham operator; Translations. mathematics, physics: differential operator used in modeling of wave propagation, heat The Spectrum of the Laplacian in Riemannian Geometry Martin Vito Cruz 25 November 2003 1 Introduction spectrum: the metric determines the Laplace operator and hence the spectrum.
On the other hand, there are only few examples of manifolds where the spectrum is known explicitly. In this section, we will examine the restrictions that The problem of the spectrum and the eigenfunctions of the LaplaceBeltrami operator on the Solmanifold Mn1 A reduces in this way to the onedimensional Schrodingers equation (5) on the line with potential Q (z).
For n 2 equation (5) reduces to the socalled modied Mathieus equation [4. Inmum of the spectrum of LaplaceBeltrami operator on a bounded pseudoconvex domain with a Kahler metric of Bergman type SongYing Li and MyAn Tran Gaps in the spectrum of LaplaceBeltrami operators up vote 2 down vote favorite Let us consider \mathbb Sd the unit Euclidean sphere of \mathbb Rd1 and let \Delta\mathbb Sd be the Laplace operator on \mathbb Sd. This more general operator goes by the name LaplaceBeltrami operator, after Laplace and Eugenio Beltrami.
Like the Laplacian, the LaplaceBeltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The spectrum will never be discrete, due to the presence of Eisenstein series in nonuniform lattices (morally, by the KazhdanMargulis theorem nonuniform lattices implies existence of unipotents). Nevertheless, by the GelfandGraev theorem, the cuspidal spectrum is always discrete!
(hence the Laplacian is essentially compact). module. Compute the LaplaceBeltrami spectrum using a linear finite element method. We follow the definitions and steps given in Reuter et al. s 2009 paper: Discrete LaplaceBeltrami Operators for Shape Analysis and Segmentation