Eigenvalue splitting for a system of Schr\'odinger operators with an energy-level crossing
| Autor: | Assal, Marouane, Fujiié, Setsuro |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schr\"odinger operators in the presence of two potential wells and with an energy-level crossing. We provide Bohr-Sommerfeld quantization condition for the eigenvalues of the system on any energy-interval above the crossing and give precise asymptotics in the semiclassical limit $h\to 0^+$. In particular, in the symmetric case, the eigenvalue splitting occurs and we prove that the splitting is of polynomial order $h^{\frac32}$ and that the main term in the asymptotics is governed by the area of the intersection of the two classically allowed domains. Comment: 27 pages, 3 figures |
| Databáze: | arXiv |
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