Result 080, Real and complex analysis

The exact Sobolev endpoint for Schrödinger convergence

Proves almost-everywhere convergence eitΔf→fe^{it\Delta}f\to f as t↓0t\downarrow0 for every f∈Hn/(2(n+1))(Rn)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 H1/3.

Proof

The bigger picture

Why it matters

How rough can a wave's initial shape be while its evolution still recovers that shape as time returns to zero? These manuscripts claim convergence at the exact critical smoothness level for the free Schrödinger equation.

What changes?

The manuscripts report convergence in every dimension n at least 2 for every initial function in H^s with s = n/(2(n+1)), including H^(1/3) in the plane. This Sobolev class consists of square-integrable functions with s derivatives of smoothness, understood fractionally. The free Schrödinger evolution returns to those initial values as positive time tends to zero at almost every spatial point, meaning outside a set of measure zero. This includes equality at the sharp smoothness threshold, not merely greater smoothness.

What does that help mathematicians do?

The evolution is defined by first removing Gaussian regularization, a smoothing procedure. The planar abstract specifies one full-measure spatial set where that limit exists for every positive time below one; the higher-dimensional abstract states that removal is uniform over this interval. Researchers can therefore recover rough initial data point by point outside a single negligible set, without needing any extra smoothness beyond the critical index.

Are there practical applications?

The immediate value is foundational: it clarifies when the free Schrödinger equation faithfully represents its initial data point by point, even at borderline regularity. This supports rigorous study of dispersive equations with rough starting profiles. The reported result is a convergence theorem, rather than a computational method or a quantitative error bound.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Endpoint convergence for the planar Schrodinger equation

September 24, 2026 83 pages

We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in H1/3(R2)H^{1/3}(\mathbb R^2), the Schrödinger evolution converges almost everywhere to the initial data as t↓0t\downarrow0. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every 0<t<10\lt t\lt 1, and the resulting evolution is continuous at t = 0.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026,
  author = {{OpenAI}},
  title = {{Endpoint convergence for the planar Schr\"odinger equation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf}{OAI:Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026}},
  year = {2026}
}

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

September 24, 2026 96 pages

We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in Hn/(2(n+1))(Rn)H^{n/(2(n+1))}(\mathbb R^n), the free Schrödinger evolution converges almost everywhere to the initial data as t↓0t\downarrow0. The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval 0<t<10\lt t\lt 1.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026,
  author = {{OpenAI}},
  title = {{Endpoint pointwise convergence for the Schr\"odinger equation in higher dimensions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026/paper.pdf}{OAI:Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.