Title: Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories Show Chinese title

Title: 半经典共振对于矩阵型薛定谔算子，在经典轨迹交叉处相互作用消失的情况

Abstract: We study the semiclassical distribution of resonances of a $2 \times 2$ matrix Schr\"odinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order $m$ of the crossings and the vanishing order $k$ of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit $h \to 0^+$, the imaginary part of the resonances is of size $h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$ when both $k$ and $m$ are odd. We also compute the first term of the associated asymptotic expansions. Show Chinese abstract

Abstract: 我们研究了半经典的共振分布，这些共振来自于一个作为分子预离解模型在Born-Oppenheimer近似下研究Hamilton量约化得到的$2 \times 2$矩阵Schrödinger算子。 所考虑的能量高于两条相关经典轨迹的能量级穿越点，并且对于这两条轨迹分别是有捕获和无捕获的。 在交叉点处的接触阶数$m$和相互作用项在交叉点处的消失阶数$k$满足某一条件时，我们证明，在半经典极限下渐近地， $h \to 0^+$时，共振的虚部大小在一般情况下为$h^{1+2(k+1)/(m+1)}$，当$k$和$m$都为奇数时收缩到$h^{1+2(k+2)/(m+1)}$。 我们还计算了相关渐近展开式的首项。