# On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits

## TL;DR

This paper investigates the conditions under which self-adjoint extensions of the Laplace-Beltrami operator on quantum circuits are invariant under the group 6, providing criteria, characterizations, and spectral analysis.

## Contribution

It introduces criteria for 6-invariant self-adjoint extensions of the Laplacian on quantum circuits and characterizes their spectral properties.

## Key findings

- Criteria for 6-invariant extensions are established.
- Explicit characterization of extensions on quantum circuits is provided.
- Spectral properties and eigenfunctions are determined for specific cases.

## Abstract

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group $G$, criteria for the existence of $G$-invariant self-adjoint extensions of the Laplace-Beltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the Laplace-Beltrami operator on an infinite set of intervals, $\Omega$, constituting a quantum circuit, which are invariant under a given action of the group $\mathbb{Z}$. A study of the different unitary representations of the group $\mathbb{Z}$ on the space of square integrable functions on $\Omega$ is performed and the corresponding $\mathbb{Z}$-invariant self-adjoint extensions of the Laplace-Beltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples.