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arXiv · 1910.01195

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

Marouane Assal·Setsuro Fujiié
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Abstract

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.

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Marouane Assal, Setsuro Fujiié. 2019-10-02. Eigenvalue splitting for a system of Schr\"odinger operators with an energy-level crossing. https://arxiv.org/abs/1910.01195

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