Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916162011398144 |
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| author | Louatron, Vincent |
| author_facet | Louatron, Vincent |
| contents | We study the semiclassical distribution of resonances of a $2 \times 2$ matrix Schrödinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order $m$ of the crossings and the vanishing order $k$ of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit $h \to 0^+$, the imaginary part of the resonances is of size $h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$ when both $k$ and $m$ are odd. We also compute the first term of the associated asymptotic expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02350 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories Louatron, Vincent Mathematical Physics Classical Analysis and ODEs We study the semiclassical distribution of resonances of a $2 \times 2$ matrix Schrödinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order $m$ of the crossings and the vanishing order $k$ of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit $h \to 0^+$, the imaginary part of the resonances is of size $h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$ when both $k$ and $m$ are odd. We also compute the first term of the associated asymptotic expansions. |
| title | Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories |
| topic | Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2306.02350 |