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

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.