Absence of localized solutions for time‑dependent positive potentials (Quantum Ping‑Pong)

Determine whether the time‑dependent Schrödinger equation i∂ψ/∂t = −Δψ + V(x,t)ψ admits no spatially localized solutions under the hypothesis that, for every fixed time t, the instantaneous Hamiltonian −Δ + V(x,t) has no bound states, in arbitrary spatial dimension.

Background

The author poses a fundamental question about localization in quantum dynamics with time‑dependent potentials. Even when each frozen‑time operator −Δ + V(x,t) has no bound states, it is unclear whether the full time‑dependent evolution can nonetheless support a localized (bound‑like) solution.

This problem is motivated both by mathematical challenges in handling time dependence (where many stationary tools fail) and by physical scenarios such as laser‑induced trapping. Adiabatic theory may treat special regimes (very slowly varying potentials), but a general resolution in any dimension is not known.

References

The broader question is: assuming the operator $-\Delta+V(x,t)$ has no bound states for each fixed $t$, does it follow that the equation $$ i\frac{\partial\psi}{\partial t}=-\Delta \psi+V(x,t)\psi $$ has NO localized solutions?

This remains an open problem in any dimension.

— A New Paradigm For Scattering Theory of Linear And Nonlinear Waves: Review And Open Problem  (2408.14269 - Soffer, 2024) in Section: Open Problems — Direct Scattering, subsection “Quantum Ping-Pong”

First-order s-points (the set $\Upsilon1_q$) appear if and only if the Hamiltonian has a discrete negative spectrum. Is this also true for s-points of order $m>1$? Does the absence of bound states ensure controllability and, consequently, the absence of s-points?

— Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system  (2609.03536 - Belishev et al., 3 Sep 2026) in Comments, bullet beginning “First-order s-points...”