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

TL;DR

This work addresses the inverse problem of determining a time‑independent Schrödinger potential from the initial‑to‑final state map on $\mathbb{R}^n$. It introduces a stationary‑state based strategy and a refined resolvent framework to handle critically singular potentials in $L^1\cap L^q$, with $q>1$ for $n=2$ and $q\ge n/2$ for $n\ge3$, avoiding complex geometrical optics. The authors prove uniqueness of the potential under these conditions, expanding prior results by permitting $L^q$ type singularities and weakening decay assumptions, especially in the time‑independent setting where no CGO solutions are needed. The methodology hinges on Alessandrini‑type orthogonality extended to stationary states and a new Banach space construction that yields decay at the endpoint, enabling the extraction of the Fourier transform of $V_1-V_2$ and thereby concluding $V_1=V_2$.

Abstract

The Schrödinger 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 $-Δ+V$ 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-Hörmander 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.

Figures (1)

• Figure 1: The exponents for the space of potentials $L^q(\mathbb{R}^n)$, the Strichartz pairs $(r,p)$, and the Stein-Tomas extension estimate $L^2(\mathbb S^{n-1})\to L^{p}(\mathbb{R}^n)$ for $p\geq q_n$. The Kenig--Ruiz--Sogge estimate is valid in the range $p\in [q_2,\infty)$ when $n=2$ and $p\in[q_n,p_n]$ when $n\geq 3$.

Theorems & Definitions (24)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Corollary 3.2
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5