The exact Sobolev endpoint for Schrödinger convergence

Claimed proofs that Schrödinger waves return to rough data at the exact smoothness threshold, in dimensions n ≥ 2.

2 papers September 24, 2026 Explained by AI

The free Schrödinger equation describes a quantum particle moving with no forces. Carleson asked how smooth the starting data must be for the solution to return to it. The return should hold at almost every point as time goes to zero. Earlier work proved convergence for every smoothness strictly above n/(2(n + 1)), and showed nothing below works. These two papers claim convergence exactly at that threshold, in the plane and in every dimension from three up.

First step

1980

Lennart Carleson, who posed the pointwise convergence problem

Significance

Major

It claims to settle the equality case of Carleson's problem, after the strict range was proved in 2017 and 2019.

Trust

Claimed, not checked

Field

Real and complex analysis

2 papers

The explanation

Start with a smooth bump on the plane, f(x) = e^(−|x|²/2). Run the free Schrödinger equation, the basic equation for a quantum particle with no forces. For this bump the solution has a closed form. At the centre its size is 1/√(1 + 4t²). At t = 0.1 that is 1/√1.04, about 0.981. As t shrinks to 0, it returns to 1, the starting value.

Rough starting data is harder. A piece of the wave with frequency N moves at speed 2N. With N = 100, it covers one unit of distance by time t = 0.005. So many fast pieces could make the solution swing wildly as t shrinks. Carleson asked how smooth f must be for the solution to return to f at almost every point. Smoothness is measured by the Sobolev space H^s, where frequency N carries a weight of about N^s.

Carleson proved convergence on the line at s = 1/4Source 1. Dahlberg and Kenig showed that this threshold is necessarySource 2. Sjölin and Vega independently got s > 1/2 in higher dimensionsSource 3. In the plane, a run of results lowered the bound to 15/32, 2/5 and then 3/8Source 4 Source 5. Bourgain's counterexample showed that s ≥ n/(2(n + 1)) is needed when n ≥ 2. It did not rule out equalitySource 6. In the plane that number is 2/(2 × 3) = 1/3. Du, Guth and Li proved planar convergence for every s > 1/3Source 7. Du and Zhang proved the strict range s > n/(2(n + 1)) for n ≥ 3Source 8.

The planar paper claims convergence at exactly s = 1/3Source 9. The second paper claims the endpoint s = n/(2(n + 1)) for every n ≥ 3Source 10. For n = 3 that is 3/8, and for n = 4 it is 2/5. Both papers say exactly how the solution is defined for rough data. They first smooth the data with a Gaussian factor and remove it, then let t go to 0Source 11. One set of full measure works for every time at onceSource 12. Full measure means it leaves out only a set of zero volume.

The main step is a maximal estimate at the exact exponentSource 13. A maximal estimate bounds the largest size of the solution over all times 0 ≤ t < 1. In the plane, the paper claims it with a weaker, local L¹ form of output. The stronger endpoint L³ maximal estimate is not claimedSource 14.

How sure is this? There is no Lean scope note for either paper, so no computer proof checker has tested them. Both papers are long and build on the Du–Guth–Li and Du–Zhang methods. No specialist review has been published.

What it doesn't show. This does not prove the stronger endpoint L³ maximal estimate in the plane, and it does not cover one dimension.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the papers hold. They settle an equality case after the strict range was known. There is no Lean scope note, and no specialist review is published.

Words used

Free Schrödinger equation: The basic equation for a quantum particle moving with no forces.
Sobolev space H^s: Functions whose frequency-N parts, weighted by about N^s, have finite total energy.
Almost everywhere: At every point except a set of zero area or volume.
Full measure: A set that leaves out only a set of zero volume.
Maximal estimate: A bound on the largest size a solution reaches over a range of times.
Endpoint: The exact threshold value of s, as opposed to every value strictly above it.

The original papers (2)

Endpoint convergence for the planar Schrodinger equation (PDF, opens in a new tab)
September 24, 2026
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions (PDF, opens in a new tab)
September 24, 2026

All quoted lines (14)

Carleson introduced the pointwise convergence problem and proved its one-dimensional positive result at regularity $s=1/4$ .
Endpoint convergence for the planar Schrodinger equation Back to text
Dahlberg and Kenig showed that this threshold is necessary.
Endpoint convergence for the planar Schrodinger equation Back to text
Sj\"olin and Vega independently obtained convergence for $s>1/2$ in higher dimensions.
Endpoint convergence for the planar Schrodinger equation Back to text
The bilinear approach of Tao and Vargas gave $s>15/32$ ; Tao's sharp bilinear restriction theorem improved this to $s>2/5$ , and Lee obtained $s>3/8$ .
Endpoint convergence for the planar Schrodinger equation Back to text
The bilinear approach of Tao and Vargas gave $s>15/32$ ; Tao's sharp bilinear restriction theorem improved this to $s>2/5$ , and Lee obtained $s>3/8$ .
Endpoint convergence for the planar Schrodinger equation Back to text
Bourgain's later counterexample forces $s\ge n/[2(n+1)]$ when $n\ge2$ , but does not rule out equality.
Endpoint convergence for the planar Schrodinger equation Back to text
Du, Guth and Li proved planar convergence and a local $L^3$ maximal estimate for $s>1/3$ .
Endpoint convergence for the planar Schrodinger equation Back to text
Du and Zhang established convergence for $s>n/[2(n+1)]$ when $n\ge3$ .
Endpoint convergence for the planar Schrodinger equation Back to text
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 $\lim_{t\downarrow0}u_f(x,t)=f^*(x).$
Endpoint convergence for the planar Schrodinger equation Back to text
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 $\lim_{a\downarrow0}\sup_{0<t<1}|U_af(x,t)-v(x,t)|=0 \qquad(x\in E_f).$
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions Back to text
The Gaussian limit is taken first.
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions Back to text
The representative of the evolution is specified by Gaussian regularization; its exceptional spatial set is independent of time.
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions Back to text
The main analytic conclusion is a local maximal estimate at the exact Sobolev exponent.
Endpoint convergence for the planar Schrodinger equation Back to text
The stronger question of an endpoint local $L^3$ maximal estimate is not asserted here.
Endpoint convergence for the planar Schrodinger equation Back to text

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