The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends

A simple sufficient condition on curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too.Comment: 25 pages, 10 figure

Water-waves modes trapped in a canal by a body with the rough surface

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter $\epsilon>0$ while the distance of the body to the water surface is also of order $\epsilon$. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely, it is proved that, for any given $d>0$ and integer $N>0$, there exists $\epsilon(d,N)>0$ such that the problem has at least $N$ eigenvalues in the interval $(0,d)$ of the continuous spectrum in the case $\epsilon\in(0,\epsilon(d,N))$. The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.Comment: 25 pages, 8 figure

Scalar boundary value problems on junctions of thin rods and plates. I. Asymptotic analysis and error estimates

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms.Comment: 34 pages, 4 figure

Exact dynamical exchange-correlation kernel of a weakly inhomogeneous electron gas

The dynamical exchange-correlation kernel $f_{xc}$ of a non-uniform electron gas is an essential input for the time-dependent density functional theory of electronic systems. The long-wavelength behavior of this kernel is known to be of the form $f_{xc}= \alpha/q^2$ where $q$ is the wave vector and $\alpha$ is a frequency-dependent coefficient. We show that in the limit of weak non-uniformity the coefficient $\alpha$ has a simple and exact expression in terms of the ground-state density and the frequency-dependent kernel of a {\it uniform} electron gas at the average density. We present an approximate evaluation of this expression for Si and discuss its implications for the theory of excitonic effects.Comment: 5 pages, 2 figure