Airy and Schrödinger-type equations on looping-edge graphs and applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908958387601408 |
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| author | Pava, Jaime Angulo Muñoz, Alexander |
| author_facet | Pava, Jaime Angulo Muñoz, Alexander |
| contents | The aim of this work is to study the Airy and Schrödinger operators on looping-edge graphs, a class of metric graphs consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. For the Airy operator, we characterize all extensions generating unitary and contractive dynamics in terms of self-orthogonal subspaces and linear operators acting on indefinite inner product spaces (Krein spaces) associated to the boundary values at the vertex. Employing similar abstract techniques, we then describe a systematic way to produce self-adjoint extensions of the Schrödinger operator that are compatible with prescribed boundary relations on looping-edge and $\mathcal{T}$-shaped graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11729 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Airy and Schrödinger-type equations on looping-edge graphs and applications Pava, Jaime Angulo Muñoz, Alexander Analysis of PDEs The aim of this work is to study the Airy and Schrödinger operators on looping-edge graphs, a class of metric graphs consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. For the Airy operator, we characterize all extensions generating unitary and contractive dynamics in terms of self-orthogonal subspaces and linear operators acting on indefinite inner product spaces (Krein spaces) associated to the boundary values at the vertex. Employing similar abstract techniques, we then describe a systematic way to produce self-adjoint extensions of the Schrödinger operator that are compatible with prescribed boundary relations on looping-edge and $\mathcal{T}$-shaped graphs. |
| title | Airy and Schrödinger-type equations on looping-edge graphs and applications |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.11729 |