It is a classical derivation that the Wigner equation, derived from the Schr\"odinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant $\epsilon\to0$. Since the latter presents the Newton's second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schr\"odinger equation. More specifically, we show that using the initial condition and final state of the Schr\"odinger equation to reconstruct the potential term, in the classical regime with $\epsilon\to0$, becomes using the initial and final state to reconstruct the potential term in the Liouville equation. This formally bridges an inverse problem in quantum mechanics with an inverse problem in classical mechanics.

翻译：典型的推论是, Wigner 等式源自包含量子信息的Schr\'odinger 等式,当重标普朗克常数 $\ epsilon\ to0$ 时,它会与Liouville 等式趋同。自后者提出牛顿第二法以来,这一过程通常被称为(半)古典界限。在本文中,我们研究Schr\'odinger 等式的反向问题的典型极限。更具体地说,我们显示,在古典制度下,利用Schr\'odinger 等式的初始条件和最终状态来重建潜在术语,用$\ epslon\\ to0$, 开始使用初始和最终状态来重建Liouville 等式中的潜在术语。这正式将量力力学的反向问题与古典机械的反向问题联系起来。