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.
Highlights
The relation between physics and symmetries has been successful and fruitful up to the point that physical theories, from the most fundamental ones, such as the Standard Model, to the effective ones applied, e.g., in condensed matter physics, are intimately related with the symmetries and transformation properties of their underlying structures, for instance, gauge symmetries in the former case or crystallographic groups in the latter.One of the aims of this article is to provide a framework for a systematic analysis of the action of symmetry groups on the configuration space of quantum circuits
Even if the configuration space is invariant under the action of a given group, not all the possible self-adjoint extensions need to be compatible with that symmetry group
After this introduction containing a critical review of previous results, we focus on the issue of self-adjoint extensions of the Laplace–Beltrami operator on quantum circuits, which are compatible with a given group of symmetries
Summary
The relation between physics and symmetries has been successful and fruitful up to the point that physical theories, from the most fundamental ones, such as the Standard Model, to the effective ones applied, e.g., in condensed matter physics, are intimately related with the symmetries and transformation properties of their underlying structures, for instance, gauge symmetries in the former case or crystallographic groups in the latter. We use the characterisation introduced in [1] to identify the set of self-adjoint extensions compatible with the action of the symmetry group in the particular case of infinite chains made up by repeating a finite block. This kind of periodic lattices is widely used in solid state physics as approximations for systems such as crystals, when the period is much smaller than the size of the system [2,3]. To have control on the symmetries that the system possess is important in the determination of the spectrum and the spaces of eigenfunctions, as they will carry the same representation The importance of this characterisation is that the space of mathematically possible self-adjoint extensions for a given quantum circuit is very large.
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