The initial-to-final-state inverse problem with critically-singular potentials

TL;DR

This paper proves that the initial-to-final-state map for the Schrödinger equation uniquely determines the potential under minimal integrability and decay conditions, extending previous results to more singular potentials.

Contribution

It establishes uniqueness of the inverse problem for time-independent Schrödinger potentials with weaker decay and singularity assumptions, using refined resolvent estimates.

Findings

Uniqueness holds for potentials in L^1 and L^q with q>1 or q≥n/2.

Refined resolvent estimates replace classical techniques.

Avoids complex geometrical optics solutions due to time-independent setting.

Abstract

The Schr\"odinger equation in high dimensions describes the evolution of a quantum system. Assume that we are given the evolution map sending each initial state $f\in L^2(\mathbb{R}^n)$ of the system to the corresponding final state at a fixed time $T$. The main question we address in this paper is whether this initial-to-final-state map uniquely determines the Hamiltonian $-\Delta+V$ that generates the evolution. We restrict attention to time-independent potentials $V$ and show that uniqueness holds provided $V \in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$, with $q>1$ if $n=2$ or $q\geq n/2$ if $n\geq 3$. This should be compared with the results of Caro and Ruiz, who proved that in the time-dependent case, uniqueness holds under the stronger assumption that the potential exhibits super-exponential decay at infinity, for both bounded and unbounded potentials. This paper extends earlier work of the same authors, where uniqueness was obtained for bounded time-independent potentials with polynomial decay at infinity. Here we only require $L^1$-type decay at infinity and allow for $L^q$-type singularities. We reach this improvement by providing a refinement of the Kenig-Ruiz-Sogge resolvent estimate, which replaces the classical Agmon-H\"ormander estimates used previously. Crucially, the time-independent setting allows us to avoid the use of complex geometrical optics solutions and thereby dispense with strong decay assumptions at infinity.