The initial-to-final-state inverse problem with critically-singular potentials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915794931154944 |
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| author | Cañizares, Manuel Caro, Pedro Parissis, Ioannis Zacharopoulos, Thanasis |
| author_facet | Cañizares, Manuel Caro, Pedro Parissis, Ioannis Zacharopoulos, Thanasis |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_12122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The initial-to-final-state inverse problem with critically-singular potentials Cañizares, Manuel Caro, Pedro Parissis, Ioannis Zacharopoulos, Thanasis Analysis of PDEs 35R30 (Primary) 35J10, 81U40 (Secondary) 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. |
| title | The initial-to-final-state inverse problem with critically-singular potentials |
| topic | Analysis of PDEs 35R30 (Primary) 35J10, 81U40 (Secondary) |
| url | https://arxiv.org/abs/2602.12122 |