The exact Sobolev endpoint for Schrödinger convergence

What OpenAI claims

Proves almost-everywhere convergence e^itΔf → f as t↓0 for every f ∈ H^n/(2(n+1))(ℝⁿ) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.

Papers (2)

1. Endpoint convergence for the planar Schrodinger equation

   24 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in H^1/3(ℝ²), the Schrödinger evolution converges almost everywhere to the initial data as t↓0. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every 0 < t < 1, and the resulting evolution is continuous at t = 0.

2. Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

   24 September 2026 · Preprint, not peer reviewed · PDF

   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))(ℝⁿ), 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.