We consider the eigenvalue problem associated to the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulas for the eigenvalues and eigenfunctions when the size of the microstructure is shorter than the wavelength of the eigenfunctions 1/ √ λ . On the other hand, it has been observed that when the size of the microstructure is of the ...

A method for solving various half-space multigroup transport problems for the case of a general transfer matrix is explained. A non-linear integral equation for the emergent distribution of the albedo problem is derived. Then, by using the full-range completeness of the infinite medium eigenfunctions, the distribution inside the half-space is obtained from the emergent distribution. Finally, th...

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel method is mathematically equivalent to isotropic heat diffu...

We discuss the existence of large isolated (non-unit) eigenvalues of the Perron– Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or ‘resonances’) are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions ...

We describe a spectral method for the direct numerical calculation of the time-dependent Schrödinger equation described in hyperspherical coordinates. The method is based on the split-step technique where the wavefunction is expanded in the appropriate eigenfunctions for the partial operators, making the time integration efficient, accurate and simple. The fast Fourier transform is applied to p...

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation length. For localized states, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact res...

We consider expansions of smooth, nonperiodic functions defined on compact intervals in eigenfunctions of polyharmonic operators equipped with homogeneous Neumann boundary conditions. Having determined asymptotic expressions for both the eigenvalues and eigenfunctions of these operators, we demonstrate how these results can be used in the efficient computation of expansions. Next, we consider t...

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichl...

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loève (KL) expansion of a GP prior. We use the Nyström approximation to obtain data dependent...