Uniform-in-ℏ bounds for Hartree wave functions

Prove uniform-in-ℏ $L^\infty$ bounds for solutions of the three-dimensional Hartree equation, specifically bounds of the form $\sup_{0<\hbar\leq 1}\sup_{t\in[0,T]}\|\psi_\hbar(t,\cdot)\|_\infty<\infty$, beyond the currently known screened-Coulomb and small-initial-data setting.

Background

The paper discusses prior semiclassical derivations for Maxwell–Schrödinger and related systems that rely on a conditional uniform-in-Planck-constant bound on the wave function. Such a bound would control the quantum density and help close stability estimates, but the paper emphasizes that it is unavailable in general.

In three dimensions, the cited literature establishes this type of estimate only for the screened Coulomb interaction under smallness assumptions on the initial data. The unresolved status of the estimate is relevant because the present work avoids assuming it by using a corrected quantum modulated energy.

References

Such bounds are not known at present, and in fact even in the simplified electrostatic settings of the Hartree equation Hartree an estimate of the form uniform in h assumption has been proved in $3D$ only in the case where $V$ is the screened Coulomb interaction and the initial data verifies smallness assumptions .

— From the Maxwell-Schrödinger equation to the Euler-Maxwell equation in the semiclassical limit  (2610.11199 - Porat, 8 Oct 2026) in Section 1, “Background”