We employ the Lippmann-Schwinger formalism to derive the analytical solutions of the transmission and reflection coefficients through a one-dimensional open quantum system, in which particle loss or gain on one lattice site located at x = 0, or particle loss and gain on the lattice sites located at \(x = \pm {\textstyle{L \over 2}}\) are considered respectively. The gain and loss on the lattice site are modeled by the delta potential with positive and negative imaginary values. The analytical solution reveals the underlying physics that the sum of the transmission and reflection coefficients through an open quantum system (even a \({\cal P}{\cal T}\)-symmetric open system) may not be 1, i.e., qualitatively explains that the number of particles is not conserved in an open quantum system. Furthermore, we find that the resonance states can be formed in the \({\cal P}{\cal T}\)-symmetric delta potential, which is similar to the case of real delta potential. The results of our analysis can be treated as the starting point of studying quantum transport problems through a non-Hermitian system using Green’s function method, and more general cases for high-dimensional systems may be deduced by the same procedure.

我们采用Lippmann-Schwinger形式主义来推导一维开放量子系统的透射系数和反射系数的解析解，在该系统中，位于x = 0的一个晶格位上的粒子损失或增益，或者位于x = 0上的一个晶格位置的粒子损失或增益。分别位于\（x = \ pm {\ textstyle {L \ over 2}} \）处的晶格位点。晶格位置上的增益和损耗由具有正和负虚数值的δ电位建模。该解析解揭示了一种基本的物理学原理，即通过一个开放量子系统（甚至是一个对称的开放系统\（{\ cal P} {\ cal T} \）的透射系数和反射系数之和可能不等于1。，即定性地解释了在开放量子系统中不守恒的粒子数量。此外，我们发现共振状态可以在\（{\ cal P} {\ cal T} \）对称的δ电位中形成，这与真实δ电位的情况相似。我们的分析结果可以作为使用格林函数方法通过非Hermitian系统研究量子输运问题的起点，并且可以通过相同的过程推导高维系统的更一般情况。