The exact Sobolev endpoint for Schrödinger convergence

Proves almost-everywhere convergence e^{it\Delta}f\to f as t\downarrow0 for every f\in H^{n/(2(n+1))}(\mathbb R^n) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.

Manuscripts (2)

Endpoint convergence for the planar Schrodinger equation

September 24, 2026 · source · cite as \cite{OAI:Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026}

We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in H^{1/3}(\mathbb R^2), the Schrödinger evolution converges almost everywhere to the initial data as t\downarrow0. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every 0\lt t\lt 1, and the resulting evolution is continuous at t = 0.

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

September 24, 2026 · source · cite as \cite{OAI:Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026}

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))}(\mathbb R^n), the free Schrödinger evolution converges almost everywhere to the initial data as t\downarrow0. The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval 0\lt t\lt 1.

Result 080 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.