Quantum lattice transport along an infinitely extended perturbation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929651715145728 |
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| author | Baradaran, Marzieh Exner, Pavel Khrabustovskyi, Andrii |
| author_facet | Baradaran, Marzieh Exner, Pavel Khrabustovskyi, Andrii |
| contents | We consider a periodic quantum graph in the form of a rectangular lattice with the $δ$-coupling of strength $γ$ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a $\widetildeγ\neγ$. We analyze the band spectrum of the system and show that it remains preserved as a set provided $\widetildeγ>γ>0$ while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals $\frac12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum lattice transport along an infinitely extended perturbation Baradaran, Marzieh Exner, Pavel Khrabustovskyi, Andrii Spectral Theory Mathematical Physics 34L05, 47A10, 81Q35, 81Q10 We consider a periodic quantum graph in the form of a rectangular lattice with the $δ$-coupling of strength $γ$ in the vertices perturbed by changing the latter at an infinite straight array of vertices to a $\widetildeγ\neγ$. We analyze the band spectrum of the system and show that it remains preserved as a set provided $\widetildeγ>γ>0$ while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals $\frac12$. |
| title | Quantum lattice transport along an infinitely extended perturbation |
| topic | Spectral Theory Mathematical Physics 34L05, 47A10, 81Q35, 81Q10 |
| url | https://arxiv.org/abs/2412.20919 |