We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

C cosh(zs) (sin z − sin x)n+1 dz, y = sinn+1 x ∮ C cosh(zs) (cos z − cos x)n+1 dz, s2 = −λ, and their associated second-order ordinary differential equationswhich are not studied are given. Complete solution is given by calculating residues for n = 1. The eigenvalues and eigenfunctions along with certain boundary conditions are explored for the first contour integral. The contour integral y(x) ...

We consider the numerical approximation of the spectrum of a secondorder elliptic eigenvalue problem by the hybridizable discontinuous Galerkin (HDG) method. We show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k + 1 and k + 1, respectively. Here k is the degree of the polynomials used to approximate the solution, its flux, an...

We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we complet...

— The paper is devote d to the finit e element analysis of second order e Hipt ie eigenvalue problems in the case when the approximate domains Oh are not subdomains of the original domain fl a U. The considérations are restricted to piecewise linear approximations and in the case of eigenfunctions to simple eigenvalues. The optimum rates of convergence for hoth the approximate eigenvalues and t...

I show that the potential V (x,m) = [b 4 −m(1−m)a(a+ 1) ] sn(x,m) dn(x,m) − b(a+ 1 2 ) cn(x,m) dn(x,m) constitutes a QES band-structure problem in one dimension. In particular, I show that for any positive integral or half-integral a, 2a+ 1 band edge eigenvalues and eigenfunctions can be obtained analytically. In the limit of m going to 0 or 1, I recover the well known results for the QES doubl...

The study of the Laplace operator and its corresponding eigenvalue problem is crucial to understand the foundations of 3D shape analysis. For that reason the most important mathematical properties of the Laplace operator in Euclidean spaces, its eigenvalues and eigenfunctions are summarized and explained in this report. The basic definitions and concepts of infinite dimensional function spaces,...

In this work we study the application of the Method of Fundamental Solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D Bilaplacian in simply connected plates. This problem was considered in [20] using wave-type functions and in [14] using radial basis function for a circular and rectangular domain. The MFS is a meshfree method that was already applied t...

Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic mechanical differential equation the presence of Dunkl operator-based deformation and investigate solutions for two important problems three-dimensional spatial space. To end, after introducing mechanics, examine Dunkl–Klein–Gordon oscillator with Cartesian spherical coord...

Some of the research surveyed here was supported by the Mathematical Sciences Research Institute in Berkeley, CA. Research at MSRI is supported in part by NSF grant DMS-9022140. We survey what is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulo...