Geometry effects in quantum dot families

Abstract

We consider Schrödinger operators in $L^2({\mathrm{R}}^{\nu }),\nu =2,3$, with the interaction in the form on an array of potential wells, each on them having rotational symmetry, arranged along a curve $\Gamma$. We prove that if $\Gamma$ is a bend or deformation of a line, being straight outside a compact, and the wells have the same arcwise distances, such an operator has a nonempty discrete spectrum. It is also shown that if $\Gamma$ is a circle, the principal eigenvalue is maximized by the arrangement in which the wells have the same angular distances. Some conjectures and open problems are also mentioned.

A few more minor improvements. Final version, to appear in Pure and Applied Functional Analysis