Semi-classical limit of an inverse problem for the Schr\"odinger equation

TL;DR

This paper investigates the semi-classical limit of an inverse problem for the Schr"odinger equation, showing that quantum inverse problems converge to classical inverse problems as the Planck constant approaches zero.

Contribution

It establishes a formal connection between quantum and classical inverse problems by analyzing the limit of potential reconstruction from initial and final states.

Findings

Quantum inverse problem converges to classical inverse problem as 

Potential reconstruction in quantum mechanics approximates classical potential recovery

Bridges quantum and classical inverse problems through semi-classical analysis

Abstract

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.