2409.10873
SPECTRAL LOCALIZATION ESTIMATES FOR ABSTRACT LINEAR SCHRÖDINGER EQUATIONS
Jingxuan Zhang
correcthigh confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s ASTLO-based proof of the spectral localization/maximal velocity bound is complete and internally consistent, culminating in Theorem 2.1 and its derivation via Theorems 2.2–2.4 and Proposition 3.4. The candidate solution tracks the same high-level strategy (smoothed cutoffs, Heisenberg dynamics, commutator expansions), but it contains a key, unjustified step: in the interaction-picture (Duhamel) representation it bounds [V(s), f(φ(t−s))] by a Lipschitz constant times ||[V(s), φ]||, which does not follow from assumption (C1) and generally requires control of ||[V(s), φ(t−s)]||. The paper avoids this pitfall by working with As(t, χ) built directly from φ and proving the needed commutator bound ‖[V(t), As(t, χ)]‖ ≤ C s^{-1}G(t). The model also incorrectly asserts integrability of ∫_0^T t^{-1}G(t) dt from G ∈ L^1 and leaves an Egorov-type refinement unproven.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The manuscript presents a clear, modular proof of spectral localization/maximal velocity bounds under natural commutator hypotheses. The ASTLO framework and recursive monotonicity (RME) yield a robust route to the target operator inequality and cleanly sidestep subtle interaction-picture pitfalls. The exposition is generally clear and well-organized, with helpful guidance on how the various components (H), (RME), (ME), and geometric comparisons assemble into Theorem 2.1. Minor improvements in cross-referencing and a brief discussion of the role of the class X would enhance clarity.