On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866917557818097664 |
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| author | Balmaseda, A. Di Cosmo, F. Pérez-Pardo, J. M. |
| author_facet | Balmaseda, A. Di Cosmo, F. Pérez-Pardo, J. M. |
| contents | 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 $\mathbb{Z}$. A study of the different unitary representations of the group $\mathbb{Z}$ on the space of square integrable functions on $Ω$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_04214 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits Balmaseda, A. Di Cosmo, F. Pérez-Pardo, J. M. Mathematical Physics 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 $\mathbb{Z}$. A study of the different unitary representations of the group $\mathbb{Z}$ on the space of square integrable functions on $Ω$ 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. |
| title | On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/1908.04214 |