On Z -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.

On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

“…with domain dom h α = {Φ ∈ H 1 ([0, 2π]) : Φ(0) = Φ(2π)}; we refer to [44][45][46] for further details.…”

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

“…These unitaries are suitable to carry representations of the symmetry groups of the underlying manifold [ILP15a]. This fact was used in [BDP19] to characterise the boundary conditions compatible with the graph structure that we consider here. Finally, let us remark that the characterisation of self-adjoint extensions that we use in this work can be used for other differential operators like Dirac operators [IP15,Pér17,Ibo+21].…”

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

“…This characterisation parametrises self-adjoint extensions in terms of unitary operators acting on the space of boundary data; these unitaries are suitable to carry representations of the symmetry groups of the underlying manifold [23]. This fact was used in [5] to characterise the boundary conditions compatible with the graph structure that we consider here.…”