Geometry effects in quantum dot families

• URL: http://arxiv.org/abs/2305.12748v3
• Date: Sun, 24 Sep 2023 08:53:59 GMT
• Title: Geometry effects in quantum dot families
• Authors: Pavel Exner
• Abstract summary: We consider Schr"odinger operators in $L2(mathrmRnu),, nu=2,3$, with the interaction in the form on an array of potential wells. 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. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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