The exact Sobolev endpoint for Schrödinger convergence

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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.

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What the papers state

Theorem 1.1 (Planar endpoint convergence)

In Endpoint convergence for the planar Schrodinger equation, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every f\in H^{1/3}(\R^2) there is a set E_f\subset\R^2 of full Lebesgue measure such that, for each x\in E_f, the finite limit

u_f(x,t)=\lim_{a\downarrow0}U_af(x,t)

exists for every real 0<t<1, and

\lim_{t\downarrow0}u_f(x,t)=f^*(x).

In fact u_f(x,\cdot) extends continuously to [0,1) with value f^*(x) at zero, and the Gaussian limit is uniform over 0<t<1.

Theorem 1.1 (Endpoint convergence)

In Endpoint pointwise convergence for the Schrodinger equation in higher dimensions, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every fixed integer n\geq3 and every complex-valued f\in H^{s_n}(\R^n), there is a full-measure set E_f of Lebesgue points of f and a function v(x,t), x\in E_f, 0\leq t<1, such that v(x,\cdot) is continuous, v(x,0)=f^*(x), and

\lim_{a\downarrow0}\sup_{0<t<1}|U_af(x,t)-v(x,t)|=0 \qquad(x\in E_f).

In particular the finite Gaussian limit

u_f(x,t)=\lim_{a\downarrow0}U_af(x,t)

exists for every real 0<t<1 on this one set, and \lim_{t\downarrow0}u_f(x,t)=f^*(x).

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Papers and explanations: Two current papers, each readable here as a page beside its PDF. One explanation by others.

Papers

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Endpoint convergence for the planar Schrodinger equation

24 September 2026, current edition.

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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

24 September 2026, current edition.

Read it here PDF

We resolve the Sobolev endpoint of Carleson’s pointwise convergence problem in every dimension n ≥ 3. For initial data in \(H^{n/(2(n+1))}(\mathbb R^n)\), 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 \(0\lt t\lt 1\).

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Explanations by others

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seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

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Within this family

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions cites Endpoint convergence for the planar Schrodinger equation.

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What people have said

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