Geometry effects in quantum dot families

TL;DR

This paper investigates how the geometry of potential well arrangements along curves in 2D and 3D affects the spectral properties of Schrödinger operators, revealing conditions for discrete spectra and eigenvalue optimization.

Contribution

It demonstrates that certain geometric configurations of potential wells induce discrete spectra and identifies arrangements that maximize principal eigenvalues, advancing understanding of geometric effects in quantum systems.

Findings

Discrete spectrum exists for bent or deformed lines with wells at equal arcwise distances.

Principal eigenvalue is maximized for wells equally spaced on a circle.

Open problems and conjectures related to spectral optimization are discussed.

Abstract

We consider Schr\"odinger 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.