论文标题
具有经典轨迹退化的系统的系统的半经典共振渐近器
Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories
论文作者
论文摘要
本文关注的是在半经典限制$ h \ to 0^+$的渐近均值$ 2 \ times 2 $matrixSchrödinger操作员在一个维度上。我们研究了两个基本的古典哈密顿轨迹在相空间中切向交叉的情况。在某些经典轨迹之一是一个简单的封闭曲线的情况下，而另一个是非捕获的，我们表明共振的虚构部分是$ h^{（M_0+3）/（M_0+1）} $，其中$ m_0 $是交叉点的最大接触顺序。该主要顺序来自交叉点的转移矩阵的次级术语，这些术语描述了微局部溶液从一个轨迹到另一个轨迹的传播。此外，我们根据与所有\ textIt {概括的经典轨迹}相关的概率幅度来明确计算共振宽度的领先系数}从封闭轨迹中逃到了无穷大。
This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schrödinger operators in one dimension. We study the case where the two underlying classical Hamiltonian trajectories cross tangentially in the phase space. In the setting that one of the classical trajectories is a simple closed curve whereas the other one is non-trapping, we show that the imaginary part of the resonances is of order $h^{(m_0+3)/(m_0+1)}$, where $m_0$ is the maximal contact order of the crossings. This principal order comes from the subprincipal terms of the transfer matrix at crossing points which describe the propagation of microlocal solutions from one trajectory to the other. In addition, we compute explicitly the leading coefficient of the resonance widths in terms of the probability amplitudes associated with all the \textit{generalized classical trajectories} escaping to infinity from the closed trajectory.
