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On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits

1
Depto. de Matemáticas, Univ. Carlos III de Madrid, Avda. de la Universidad 30, Leganés, 28911 Madrid, Spain
2
Instituto de Ciencias Matemáticas (CSIC - UAM - UC3M - UCM) ICMAT, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 1047; https://doi.org/10.3390/sym11081047
Received: 16 July 2019 / Revised: 6 August 2019 / Accepted: 10 August 2019 / Published: 14 August 2019
(This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics)
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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. View Full-Text
Keywords: groups of symmetry; self-adjoint extensions; quantum circuits groups of symmetry; self-adjoint extensions; quantum circuits
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Balmaseda, A.; Di Cosmo, F.; Pérez-Pardo, J.M. On Z -Invariant Self-Adjoint Extensions of the Laplacian on Quantum Circuits. Symmetry 2019, 11, 1047.

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