OpenAI · 080-02 · 2026-09-24 · Real and complex analysis

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026

Family 080. The Sobolev endpoint in Carleson's Schrödinger convergence problem.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves almost-everywhere convergence $e^{i t Δ} f \to f$ as $t ↓ 0$ for every $f \in H^{n / (2 (n + 1))} (R^{n})$ and every dimension $n \geq 2$ . This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint $H^{1/3}$ .

We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in $H^{n / (2 (n + 1))} (R^{n})$ , the free Schrödinger evolution converges almost everywhere to the initial data as $t ↓ 0$ . The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval $0 < t < 1$ .

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.