In this paper, we study an eigenvalue problem for stochastic Hamiltonian systems driven by a Brownian motion and Poisson process with boundary conditions. By means of dual transformation and generalized Riccati equation systems, we prove the existence of eigenvalues and construct the corresponding eigenfunctions. Moreover, a specific numerical example is considered to illustrate the phenomenon ...

This paper considers the problem of viscous dissipation in the flow of Newtonian fluid through a tube of annular cross section, with Dirichlet boundary conditions. The solution of the problem is obtained by a series expansion about the complete eigenfunctions system of a Sturm-Liouville problem. Eigenfunctions and eigenvalues of this Sturm-Liouville problem are obtained by Galerkin’s method.

Vibrational eigenfunctions are calculated on-the-fly using semiclassical methods in conjunction with ab initio density functional theory classical trajectories. Various semiclassical approximations based on the time-dependent representation of the eigenfunctions are tested on an analytical potential describing the chemisorption of CO on Cu(100). Then, first principles semiclassical vibrational ...

Consider a family of bounded domains Ωt in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian ∆t on each of them. Then we can organize all the eigenfunctions into continuous families u (j) t with eigenvalues λ (j) t also varying continuously with t, although the relative sizes of the eigenvalues will change ...

This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and L norms of global eigenfunctions. The emphasis is on the connection between the behavio...

Given a family X(t) of continuous–time nearest–neighbor random walks on the one dimensional lattice Z, parameterized by n ∈ N+, we show that the spectral analysis of the Markov generator of X with Dirichlet conditions outside (0, n) reduces to the analysis of the eigenvalues and eigenfunctions of a suitable generalized second order differential operator −DmnDx with Dirichlet conditions outside ...

It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states—the steady coalescent pole solutions. In previo...

I revisit the so called “bispectral problem” introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued.

In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators (and one dimensional) a much simpler the...

In this paper we investigate the semiclassical behavior of the lowest eigenvalues of a model Schrödinger operator with variable magnetic field. This work aims at proving an accurate asymptotic expansion for these eigenvalues, the corresponding upper bound being already proved in the general case. The present work also aims at establishing localization estimates for the attached eigenfunctions. ...