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The initial-to-final-state inverse problem with unbounded potentials and Strichartz estimates

Pedro Caro, Alberto Ruiz

TL;DR

This work extends the initial-to-final-state inverse problem for the Schrödinger equation to unbounded potentials by establishing uniqueness under precise time-space decay, leveraging a full suite of Strichartz estimates including the endpoint. A central technical achievement is a robust CGO construction with a sharp remainder bound, achieved via a Lavine–Nachman trick that reduces to a contraction of a Birman–Schwinger operator for large frequencies. The paper also clarifies the limitations of Bourgain-space approaches in this setting by constructing counterexamples showing non-embedding into mixed-norm Strichartz spaces, and it juxtaposes these results with the Calderón problem. Together, these results highlight the critical role of time-dependent Strichartz bounds and the limitations of Bourgain-space methods in data-driven quantum inverse problems with unbounded interactions.

Abstract

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. We formulated this problem to establish a theoretical framework that would explain the viability of data-driven prediction in quantum mechanics. In a previous work, we analysed this inverse problem for Hamiltonians of the form $-Δ+ V$ with an electric potential $V = V({\rm t}, {\rm x})$, and we showed that uniqueness holds whenever the potentials are bounded and decay super-exponentially at infinity. In this paper, we extend this result for unbounded potentials. One of the key steps consists in proving a family of suitable Strichartz estimates -- including the corresponding endpoint of Keel and Tao. In the context of the inverse Calderón problem this family of inequalities corresponds to the Carleman inequality proved by Kenig, Ruiz and Sogge. Haberman showed that this inequality can be also retrieved as an embedding of a suitable Bourgain space. The corresponding Bourgain space in our context do not capture the mixed-norm Lebesgue spaces of Strichartz inequalities. In this paper, we give a counterexample that justifies this fact, and shows the limitations of Bourgain spaces to address the initial-to-final-state inverse problem.

The initial-to-final-state inverse problem with unbounded potentials and Strichartz estimates

TL;DR

This work extends the initial-to-final-state inverse problem for the Schrödinger equation to unbounded potentials by establishing uniqueness under precise time-space decay, leveraging a full suite of Strichartz estimates including the endpoint. A central technical achievement is a robust CGO construction with a sharp remainder bound, achieved via a Lavine–Nachman trick that reduces to a contraction of a Birman–Schwinger operator for large frequencies. The paper also clarifies the limitations of Bourgain-space approaches in this setting by constructing counterexamples showing non-embedding into mixed-norm Strichartz spaces, and it juxtaposes these results with the Calderón problem. Together, these results highlight the critical role of time-dependent Strichartz bounds and the limitations of Bourgain-space methods in data-driven quantum inverse problems with unbounded interactions.

Abstract

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. We formulated this problem to establish a theoretical framework that would explain the viability of data-driven prediction in quantum mechanics. In a previous work, we analysed this inverse problem for Hamiltonians of the form with an electric potential , and we showed that uniqueness holds whenever the potentials are bounded and decay super-exponentially at infinity. In this paper, we extend this result for unbounded potentials. One of the key steps consists in proving a family of suitable Strichartz estimates -- including the corresponding endpoint of Keel and Tao. In the context of the inverse Calderón problem this family of inequalities corresponds to the Carleman inequality proved by Kenig, Ruiz and Sogge. Haberman showed that this inequality can be also retrieved as an embedding of a suitable Bourgain space. The corresponding Bourgain space in our context do not capture the mixed-norm Lebesgue spaces of Strichartz inequalities. In this paper, we give a counterexample that justifies this fact, and shows the limitations of Bourgain spaces to address the initial-to-final-state inverse problem.

Paper Structure

This paper contains 25 sections, 21 theorems, 424 equations.

Table of Contents

  1. Introduction
  2. General scheme to prove the \ref{['th:uniqueness']}
  3. Bourgain spaces for the initial-to-final-state inverse problem
  4. Contents
  5. Complex geometrical optics solutions
  6. Preliminary discussion
  7. The Lavine--Nachman trick
  8. Boundedness properties of $S_\nu$
  9. Proof of the \ref{['th:gain_Snu']}
  10. Proof of the \ref{['th:non-gain_Snu']}
  11. Bounds for the Birman--Schwinger operator
  12. Hamiltonians with identical initial-to-final-state maps
  13. An integral identity for physical solutions
  14. Dual form of the orthogonality relation
  15. An integration by parts for exponential growing functions
  16. ...and 10 more sections

Key Result

Theorem 1

Consider $V_1$ and $V_2$ in $L^a((0, T); L^b (\mathbb{R}^n))$ with $n \geq 2$ and $(a, b) \in [1, \infty] \times [1, \infty]$ satisfying For the endpoint $(a, b) = (\infty, n/2)$ with $n \geq 3$, we also suppose that $V_1$ and $V_2$ belongs to $C ([0, T]; L^{n/2}(\mathbb{R}^n))$. Let $\mathcal{U}_T^1$ and $\mathcal{U}_T^2$ denote the initial-to-final-state map associated to the Hamiltonians $-\De

Theorems & Definitions (48)

  • Theorem 1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of the \ref{['prop:LNtrick']}
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • ...and 38 more