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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.01995 |
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| _version_ | 1866913458318999552 |
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| author | Pistol, Mats-Erik Pivovarchik, Vyacheslav |
| author_facet | Pistol, Mats-Erik Pivovarchik, Vyacheslav |
| contents | Let v_1 and v_2 be two distinct vertices of a tree T_0. Let ϕ_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let ϕ_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then ϕ_N(λ)^{(1)}\equiv ϕ_N(λ)^{(2)} and ϕ_D(λ)^{(1)}\equiv ϕ_D(λ)^{(1)} (which means that the scattering is the same at v_1 and v_2). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cospectral trees indistinguishable by scattering Pistol, Mats-Erik Pivovarchik, Vyacheslav Mathematical Physics 34B45, 34B24, 34L20 Let v_1 and v_2 be two distinct vertices of a tree T_0. Let ϕ_N^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 rooted at v_i with Neumann conditions at the root and let ϕ_D^{(i)} (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T_0 with Dirichlet conditions at the root. We prove that if attaching any tree to T_0 at the vertices v_1 and v_2 leads to cospectral trees and d(v_1)=d(v_2) then ϕ_N(λ)^{(1)}\equiv ϕ_N(λ)^{(2)} and ϕ_D(λ)^{(1)}\equiv ϕ_D(λ)^{(1)} (which means that the scattering is the same at v_1 and v_2). |
| title | Cospectral trees indistinguishable by scattering |
| topic | Mathematical Physics 34B45, 34B24, 34L20 |
| url | https://arxiv.org/abs/2408.01995 |