Stability estimates for the initial-to-final-state inverse problem
Abstract
The initial-to-final-state inverse problem for the Schrödinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state f∈ L2(ℝn) to the corresponding final state at a fixed time T. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove Hölder stability estimates.
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BibTeX
@misc{canizares2026stability,
title = {{Stability estimates for the initial-to-final-state inverse problem}},
author = {Manuel Cañizares and Thanasis Zacharopoulos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00257},
eprint = {2609.00257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00257},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stability estimates for the initial-to-final-state inverse problem
AU - Manuel Cañizares
AU - Thanasis Zacharopoulos
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00257
UR - https://arxiv.org/abs/2609.00257
AB - The initial-to-final-state inverse problem for the Schrödinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state f∈ L^2(ℝ^n) to the corresponding final state at a fixed time T. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove Hölder stability estimates.
ER -