Can bounded positive time-dependent potentials trap a Schrödinger wave?
Consider potentials $V \in C^\infty([0, \infty) \times \mathbb{R}^3; \mathbb{R})$ with $V \ge 0$, every space-time derivative of $V$ bounded, and $\operatorname{supp} V(t, \cdot) \subset B_R$ for all $t \ge 0$, for one fixed $R < \infty$. For $\psi_0 \in L^2(\mathbb{R}^3)$ let $\psi$ be the solution, given by the unitary propagator, of $$i \partial_t \psi = \bigl(-\Delta + V(t, x)\bigr) \psi, \qquad \psi(0) = \psi_0.$$ Soffer, who calls the question quantum ping-pong, notes that such a potential can have bound states only if it depends on time, and asks more broadly whether this equation can have localized solutions when $-\Delta + V(t, \cdot)$ has no bound states for each fixed $t$.
Problem. Is there such a potential $V$, together with $\psi_0 \in L^2(\mathbb{R}^3)$ of norm one, such that the solution is uniformly localized in the sense that $$\lim_{L \to \infty} \sup_{t \ge 0} \int_{|x| > L} |\psi(t, x)|^2 \, dx = 0?$$ The potential may vary arbitrarily in time within these bounds. A negative answer would say that no nonzero state stays spatially tight under any potential in this class.
Source: Soffer (2024), §14.2, "Quantum ping-pong": whether a positive, bounded, regular potential localized in space can have bound states. Stated in this form, in three dimensions, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, bound states, Schrödinger equations, time-dependent potentials
References
@article{soffer2024,
author = {Soffer, Avy},
title = {A new paradigm for scattering theory of linear and nonlinear waves: review and open problems},
journal = {Adv. Contin. Discrete Models},
volume = {2024},
year = {2024},
pages = {Paper No. 34},
doi = {10.1186/s13662-024-03831-6},
eprint = {2408.14269},
archivePrefix = {arXiv},
}
@misc{aim-positive-time-dependent-potential-localization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Can bounded positive time-dependent potentials trap a Schrödinger wave?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/445-positive-time-dependent-potential-localization.md},
note = {Entry 445 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00463, "Can bounded positive time-dependent potentials trap a Schrödinger wave?", clopen.solutions.