An approach to spectral problems on Riemannian manifolds (original) (raw)
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1 an Approach to Spectral Problems on Riemannian Manifolds
2016
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.
A generalized expansion method for computing Laplace-Beltrami eigenfunctions on manifolds
arXiv (Cornell University), 2022
Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a smooth bounded domain based on the relaxation to the Schrödinger operator with finite potential on a Riemannian manifold and projection in a special basis. We prove spectral exactness of the method and provide examples of calculated results and applications, particularly, in quantum billiards on manifolds.
Analytic and geometric aspects of Laplace operator on Riemannian manifold
Malaya Journal of Matematik
In the past decade there has been a flurry of work at intersection of spectral theory and Riemannian geometry. In this paper we present some of recent results on abstract spectral theory depending on Laplace-Beltrami operator on compact Riemannian manifold. Also, we will emphasize the interplay between spectrum of operator and geometry of manifolds by discussing two main problems (direct and inverse problems) with an eye towards recent developments.
The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds
Duke Mathematical Journal, 2001
Given a complete Riemannian manifold M and a Schrödinger operator − + m acting on L p (M), we study two related problems on the spectrum of − +m. The first one concerns the positivity of the L 2-spectral lower bound s(− + m). We prove that if M satisfies L 2-Poincaré inequalities and a local doubling property, then s(− + m) > 0, provided that m satisfies the mean condition inf p∈M 1 |B(p, r)| B(p,r) m(x) d x > 0 for some r > 0. We also show that this condition is necessary under some additional geometrical assumptions on M. The second problem concerns the existence of an L p-principal eigenvalue, that is, a constant λ ≥ 0 such that the eigenvalue problem u = λmu has a positive solution u ∈ L p (M). We give conditions in terms of the growth of the potential m and the geometry of the manifold M which imply the existence of L p-principal eigenvalues. Finally, we show other results in the cases of recurrent and compact manifolds.
Comptes Rendus Mathematique, 2013
We study the discreteness of the spectrum of Schrödinger operators which are defined on N -dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N -dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N -dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.
The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory
Journal of Functional Analysis, 1987
Using techniques of stationary scattering theory for the Schrodinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds M" arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds M" of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements. 0 1987 Academic press, hc.
Eigenvalue estimates for the -Laplace operator on manifolds
Nonlinear Analysis: Theory, Methods & Applications, 2010
We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of the manifold.
Eigenvalues of polyharmonic operators on variable domains
ESAIM: Control, Optimisation and Calculus of Variations, 2013
We consider a class of eigenvalue problems for poly-harmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains.
Remarks on a polyharmonic eigenvalue problem
Comptes Rendus Mathematique, 2010
Presented by Philippe G. Ciarlet This Note deals with a nonlinear eigenvalue problem involving the polyharmonic operator on a ball in R n . The main result of this Note establishes the existence of a continuous spectrum of eigenvalues such that the least eigenvalue is isolated. r é s u m é On considère un problème non linéaire de valeurs propres associé à l'opérateur polyharmonique sur une boule dans R n . Dans cette Note on montre l'existence d'un spectre continu de valeurs propres tel que la valeur propre principale est isolée. Soit B une boule de rayon R > 0 dans R n et soit K un entier strictement positif. Dans cette Note on étudie le problème non linéaire de valeurs propres