Abstract. We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We show that if Ω0 ⊂ Ωǫ are bounded domains (although not necessarily uniformly bounded) and we know that the eigenvalues and eigenfunctions with Neumann boundary condition in Ωǫ converge to the ones in Ω0, then necessarily we...

We consider the Dirichlet Laplacian ∆ in a family of bounded domains {−a < x < b, 0 < y < h(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulae obtained involve the eigenvalues and e...

Extended Abstract. When data are in the form of continuous functions, they may challenge classical methods of data analysis based on arguments in finite dimensional spaces, and therefore need theoretical justification. Infinite dimensionality of spaces that data belong to, leads to major statistical methodologies and new insights for analyzing them, which is called functional data analysis (FDA...

In this work we analyze the boundary value problems on a path associated with Schrödinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green’s function for each regular problem and th...

Series expansions are obtained for a rich subset of eigenvalues and eigenfunctions of an operator that arises in the study of rectangular membranes: the operator is the 2-D Laplacian with restorative force term and Dirichlet boundary conditions. Expansions are extracted by considering the restorative force term as a linear perturbation of the Laplacian; errors of truncation for these expansions...

Abstract: In this paper, we present a novel method for computation of eigenvalues and eigenfunctions for a class of singular Sturm-Liouville boundary value problems using modified Adomian decomposition method. The proposed method can be applied to any type of regular as well as singular Sturm-Liouville problems. This current method is capable of finding any n-th eigenvalues and eigenfunctions o...

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU (2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

The subject of this talk concerns to the classical inverse spectral problem. This inverse problem can be formulated as follows: do the Dirichlet eigenvalues and the derivatives (which order?) of the normalized eigenfunctions at the boundary determine uniquely the coefficients of the corresponding differential operator? For operators of order two this type of theorem is called Borg-Levinson theo...

Abstract The toroidal dipole represents the lowest order in family of multipoles. They appear physics at all scales, from particle to condensed matter physics, and metamaterials. Nevertheless, small systems they have be investigated context quantum mechanics operators are introduced for projections on Cartesian axes. Here we give analytical expressions eigenvalues generalized eigenfunctions <?C...

The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman’s method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained. 1. After the remarkable papers of T. Carleman [1–2] the method based on the asymptotic investigation of the r...