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

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 $-\Delta + V$ 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 Calder\'on 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.