We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Focusing on the cases where the two underlying classical periodic trajectories cross to each other, we give Bohr–Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consist in the eigenvalue splitting which occurs when the two action integrals along the closed trajectories coincide. The splitting is of polynomial order \(h^{{\frac{4}{3}}}\) in the tangential case and of order \(h^{{\frac{3}{2}}}\) in the transversal case, and the coefficients of the leading terms reflect the geometry of the crossing.

我们研究耦合薛定谔算子的一维二乘二半经典系统的特征值的渐近分布，其中每个算子都具有简单的势阱。关注两个基本的经典周期轨迹相互交叉的情况，我们在切向和横向交叉情况下为系统的特征值给出了 Bohr-Sommerfeld 类型的量化规则。我们的主要结果包括当沿着闭合轨迹的两个动作积分重合时发生的特征值分裂。分裂是多项式的\(h^{{\frac{4}{3}}}\)在切线的情况下和阶\(h^{{\frac{3}{2}}}\)在横向情况，主要项的系数反映了交叉的几何形状。