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

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.