Eigenvalue splitting for a system of Schrödinger operators with an energy-level crossing

Abstract

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrödinger 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 $h^{\frac{3}{2}}$ and that the main term in the asymptotics is governed by the area of the intersection of the two classically allowed domains.

27 pages, 3 figures

References in corpus (2)

• Bohr-Sommerfeld phases for avoided crossings
• Widths of resonances above an energy-level crossing