On <inline-formula> <mml:math id="mm3333" display="block"> <mml:semantics> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:semantics> </mml:math> </inline-formula>-Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits

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, Ω , constituting a quantum circuit, which are invariant under a given action of the group Z . A study of the different unitary representations of the group Z on the space of square integrable functions on Ω is performed and the corresponding 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.

Keywords

• groups of symmetry
• self-adjoint extensions
• quantum circuits