• Corpus ID: 203642101

Eigenvalue splitting for a system of Schr\"odinger operators with an energy-level crossing.

@article{Assal2019EigenvalueSF,
  title={Eigenvalue splitting for a system of Schr\"odinger operators with an energy-level crossing.},
  author={Marouane Assal and Setsuro Fujii'e},
  journal={arXiv: Mathematical Physics},
  year={2019}
}

• M. Assal, Setsuro Fujii'e
• Published 2 October 2019
• Mathematics
• arXiv: Mathematical Physics

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrodinger 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.