具有重叠支集的位势转移矩阵的合成
Composition of Transfer Matrices for Potentials with Overlapping Support
摘要：
对于一对具有支撑$I_j$和转移矩阵$M_j$的实散射势或复散射势$v_j:\mathbb{R}\to\mathbb{C}$（$j=1,2$），如果$I_1$位于$I_2$的左侧，则$v_1+v_2$的转移矩阵由乘积$M_2 M_1$给出。我们探讨的$I_1$和$I_2$有一个小的交集的情况下，推广这种组合规则的前景。特别是，我们表明，如果$I_1$和$I_2$相交于一个有限的封闭区间长度$\ell$，其中两个潜在的是解析的，那么最低阶的修正上述组成规则是成比例的$\ell^5$。如果$v_1$和$v_2$在除$x=\ell$和$x=0$外的整个时间间隔内均为分析型，则该校正为$\ell^3$级。我们利用这些结果来探讨叠加的一对单向不可见的潜力重叠的支持。
英文摘要：
For a pair of real or complex scattering potentials $v_j:\mathbb{R}\to\mathbb{C}$ ($j=1,2$) with support $I_j$ and transfer matrix $M_j$, the transfer matrix of $v_1+v_2$ is given by the product $M_2 M_1$ provided that $I_1$ lies to the left of $I_2$. We explore the prospects of generalizing this composition rule for the cases that $I_1$ and $I_2$ have a small intersection. In particular, we show that if $I_1$ and $I_2$ intersect in a finite closed interval of length $\ell$ in which both the potentials are analytic, then the lowest order correction to the above composition rule is proportional to $\ell^5$. This correction is of the order of $\ell^3$, if $v_1$ and $v_2$ are respectively analytic throughout this interval except at $x=\ell$ and $x=0$. We use these results to explore the superposition of a pair of unidirectionally invisible potentials with overlapping support.
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