The exact Sobolev endpoint for Schrödinger convergence

1 · The essentials

The Schrödinger equation describes how an initial function changes with time. This family of manuscripts asks whether that changing function returns to its starting value at almost every location as time approaches zero. The answer claimed is yes when the initial function meets an exact requirement on its smoothness. In two dimensions, that requirement is membership in the space $H^{1/3}$ . The companion manuscript gives the corresponding requirement in every higher dimension. Both manuscripts specify how to define the evolving function by first removing a Gaussian smoothing procedure, before letting time approach zero. These results are claimed in the manuscripts; the main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine watching a changing picture at times ever closer to its opening frame, asking whether the value at almost every location returns to its original value. The manuscripts also require a particular order of operations: remove a smoothing filter first, then move time toward zero.

Why it matters

The repository describes these claims as attaining the sharp Sobolev equality case of Carleson's Schrödinger convergence problem. The focus is the exact smoothness threshold itself, including $H^{1/3}$ in the plane, rather than a requirement of additional smoothness.

2 · Going further

The precise claim is that, for every dimension $n \geq 2$ and every $f \in H^{n / (2 (n + 1))} (R^{n})$ , the free Schrödinger evolution satisfies $e^{i t Δ} f (x) \to f (x)$ as $t ↓ 0$ for almost every $x$ . Here, “almost every” permits an exceptional spatial set of measure zero. The repository identifies this as the sharp Sobolev equality case of Carleson's convergence problem.

The planar manuscript treats $n = 2$ , where the exponent is $1/3$ . Its abstract specifies that the evolution is defined by taking the Gaussian regularization limit first. On one full-measure spatial set, that limit exists for every $0 < t < 1$ , and the resulting evolution is continuous at $t = 0$ with the initial value.

The companion manuscript treats every $n \geq 3$ with exponent $n / (2 (n + 1))$ . Its abstract describes removing Gaussian regularization on one full-measure spatial set, uniformly over $0 < t < 1$ . Together, the two manuscripts cover the dimensions stated in the repository summary.

The abstracts identify the regularization construction and the shared full-measure set as features of the claims, but they do not describe the estimates or argument establishing convergence. Thus, the supplied text supports an account of the precise conclusions and their construction, not a reconstruction of the mathematical argument. The main result is not formalized in Lean and has not been formally verified.

Useful notions

Functions on Euclidean space
Limits as time approaches zero
Measure-zero exceptional sets
Sobolev smoothness
Gaussian regularization

What was known before / what is new

Before

The source places the results within Carleson's Schrödinger convergence problem and describes the target as a sharp Sobolev endpoint. It gives no account of earlier estimates, counterexamples, or previously settled dimensions, so no more detailed history can be inferred from it.

New

The planar manuscript claims almost-everywhere convergence for every initial function in $H^{1/3} (R^{2})$ at the exact endpoint. The companion manuscript claims the corresponding endpoint result in every dimension $n \geq 3$ . Both specify a Gaussian-regularization construction on a single full-measure spatial set, with the higher-dimensional abstract explicitly stating uniform removal over $0 < t < 1$ .

Limits and assumptions

The claims require initial data in $H^{n / (2 (n + 1))} (R^{n})$ and concern the free Schrödinger evolution in dimensions $n \geq 2$ . Convergence is almost everywhere, not necessarily at every spatial point, and the evolution is defined by removing Gaussian regularization before taking $t ↓ 0$ . The supplied abstracts do not give the argument or state conclusions for lower smoothness, dimension one, or other equations. The main result is not formalized in Lean and has not been formally verified. The supplied text does not identify further open questions.

Mini glossary

Sobolev space $H^{s}$: A function space expressing a smoothness requirement indexed by the exponent $s$ .
Endpoint: The exact equality case of the smoothness threshold under discussion.
Almost-everywhere convergence: Convergence at every spatial point except possibly those in a set of measure zero.
Full-measure spatial set: A set of spatial points whose complement has measure zero.
Gaussian regularization: The Gaussian smoothing procedure whose removal defines the evolution in these manuscripts.
Schrödinger evolution: The time-dependent function denoted by $e^{i t Δ} f$ that evolves from the initial function $f$ .

Keywords : Schrödinger convergence · Sobolev endpoint · Gaussian regularization

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Endpoint convergence for the planar Schrodinger equation

24/09/2026

To be verified

We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in $H^{1/3} (R^{2})$ , 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.

PDF GitHub folder

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

24/09/2026

To be verified

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 $0 < t < 1$ .

PDF GitHub folder