标题： 多维相空间隧穿中的共振宽度 显示英文标题

标题： Resonance widths in a case of multidimensional phase space tunneling

摘要： 我们考虑一个半经典$2\times 2$矩阵薛定谔算子，形式为$P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$，其中$\mu$和$\tau$是两个小的正常数，$V_2$是实解析的并在 0 处具有非退化的最小值，$R=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$是一个对角线外的$2\times 2$对称一阶微分算子矩阵，其系数是解析的。 然后，记$e_1$为$-\Delta + \la \tau V_2"(0)x,x\ra /2$的第一个特征值，并在$r_{1,2}=r_{2,1}^*$上满足某种椭圆性条件的情况下，我们证明，对于任何足够小的$\mu$，以及对于具有某些$\tau(\mu)>0$的$0<\tau \leq\tau(\mu)$，唯一共振$\rho$的$P$使得$\rho = \tau V_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$（当$h\rightarrow 0_+$时）满足，$$ \Im \rho = -h^{\frac32}f(h,\ln\frac1{h})e^{-2S/h}, $$其中$f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m$是一个符号，$f_{0,0}>0$，而$S$是沿包含$(0,0)$并由$p_1:=\xi^2 - x_n-\mu$和$p_2:=\xi^2 +\tau V_2(x)$的复数零双特征线的并集组成的方便闭合路径上的复数作用量的虚部（断裂瞬子）。 这种断裂的瞬子是通过与$p_2$在点$(0,0)$相关的出射和入射复数拉格朗日流形来描述的，以及它们与$p_1$的特征集$p_1^{-1}(0)$的交点。 显示英文摘要

摘要： We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and admits a non degenerate minimum at 0, and $R=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order differential operators with analytic coefficients. Then, denoting by $e_1$ the first eigenvalue of $-\Delta + \la \tau V_2"(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}=r_{2,1}^*$, we show that, for any $\mu$ sufficiently small, and for $0<\tau \leq\tau(\mu)$ with some $\tau(\mu)>0$, the unique resonance $\rho$ of $P$ such that $\rho = \tau V_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$ (as $h\rightarrow 0_+$) satisfies, $$ \Im \rho = -h^{\frac32}f(h,\ln\frac1{h})e^{-2S/h}, $$ where $f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m$ is a symbol with $f_{0,0}>0$, and $S$ is the imaginary part of the complex action along some convenient closed path containing $(0,0)$ and consisting of a union of complex nul-bicharacteristics of $p_1:=\xi^2 - x_n-\mu$ and $p_2:=\xi^2 +\tau V_2(x)$ (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with $p_2$ at the point $(0,0)$, and their intersections with the characteristic set $p_1^{-1}(0)$ of $p_1$.