Filtration de certains espaces de fonctions sur un espace symétrique réductif

We introduce a filtration of a -module of some space of functions on a reductive symmetric space G/H, and compute the associated grading as a direct sum of induced representations. As an application of this result to the reductive groups viewed as symmetric spaces, we are able to realize any Harish-Chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups, the inducing representations being trivial on the unipotent radical.     

Spectral theory of left definite difference operators

This paper is concerned with the spectral theory for the second-order left definite difference boundary value problems. Existence of eigenvalues of boundary value problems is proved, numbers of their eigenvalues are calculated and fundamental spectral results are obtained.     

紧致对称空间的热核、特征值和特征函数

本文利用群表示论研究李群以及紧致对称空间的热核，特征值与特征函数．特别讨论了复格拉斯曼流形以及流形Sp（n）／U（n）上特征值及特征函数。     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz,analysis, FEM approximations and computations

In this work we focus on the efficient representation and computation of the eigenvalues and eigenfunctions of the surface Schrödinger operator ${(i\mathrm{\nabla }+{\mathbf{A}}_0)}^2$ that governs a class of nonlinear Ginzburg–Landau (GL) superconductivity models on rotationally symmetric Riemannian 2-manifolds S. We identify and analyze a complete orthonormal system in $L^2(S;\mathbb{C})$ of eigenmodes having a variable-separated form. For the unknown functions in this ansatz, our analysis facilitates the identification of approximate spectral problems whose eigenvalues lie arbitrarily near corresponding eigenvalues of the Schrödinger operator. We then develop and implement an arbitrary order finite element method for the efficient numerical approximation of the eigenvalue problem. We also demonstrate our analysis, algorithm and its convergence rate using parallel computations performed on a variety of choices of smooth and non-smooth surfaces S.     

A perturbed Whittaker-Kotel?nikov-Shannon sampling theorem

The sampling theorem of Whittaker (1915) [31], Kotel?nikov (1933) [25] and Shannon (1949) [28] gives cardinal series representations for finite L²-Fourier transforms at equidistant sampling points. Here we investigate the situation when the Fourier transform is replaced by a perturbed one. Thus the kernel of the transform will be of the form exp(−ixt)+ε(x,t), instead of exp(−ixt) in the unperturbed case. The perturbed kernel arises from first order eigenvalue problems with rank one perturbations.     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.