On the critical exponent in an isoperimetric inequality for chords

The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for $1 \le p \le 2$, but this is not the case for sufficiently large values of $p$. Here we determine the critical value $p_c(u)$ of $p$ above which the circle is not a local maximizer finding, in particular, that $p_c(\frac12 L)=\frac52$. This corrects a claim made in \cite{EHL}.Comment: LaTeX, 12 pages, with 1 eps figur

Spectral estimates for Dirichlet Laplacians and Schroedinger operators on geometrically nontrivial cusps

The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schr\"odinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.Comment: LaTeX, 18 pages; minor improvement, references added; to appear in Journal of Spectral Theor

Approximations by graphs and emergence of global structures

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.Comment: 6 pages, RevTeX with 5 ps figure

Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide

In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of $\delta$-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.Comment: LaTeX, 16 page