Let h_1(t)=Cte^-pi t^2 be the first Hermite function, where C≠ 0, and for a,b>0 define the Gabor system G(h_1,a,b)=e^2 pi i mbth_1(t-na):m,n∈Z. Determine whether G(h_1,a,b) is a frame for L^2(R) exactly when ab 0.

Lyubarskii–Nes conjecture on the frame set of the first Hermite function

Let $h_1(t)=Cte^{-pi t^2}$ be the first Hermite function, where $C\neq 0$ , and for $a,b>0$ define the Gabor system $\mathcal{G}(h_1,a,b)=\{e^{2\pi i mbt}h_1(t-na):m,n\in\mathbb{Z}\}$ . Determine whether $\mathcal{G}(h_1,a,b)$ is a frame for $L^2(\mathbb{R})$ exactly when $ab<1$ and $ab\neq N/(N+1)$ for every positive integer $N$ . Equivalently, the conjecture asserts that the non-frame set is $\{(a,b)\in(0,\infty)^2:ab\geq 1\}\cup\{(a,b)\in(0,\infty)^2:ab=N/(N+1)\text{ for some }N\in\mathbb{Z}_{>0}\}$ .

References

Primary source

arXiv

Additional references

The frame set of the first Hermite function — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed September 2026 preprint claims to settle the conjecture, but the result has not yet been independently verified.

The Lyubarskii–Nes conjecture predicts that the first Hermite function generates a Gabor frame exactly for lattice volume less than $1$ , except at volumes $N/(N+1)$ for positive integers $N$ .

Known results

Lyubarskii and Nes proved the rectangular-lattice frame result for the first Hermite function and odd functions.
For rectangular lattices, framing is known when $ab<1/2$ .
Non-framing is known at $ab=N/(N+1)$ for every positive integer $N$ .
The region $1/2<ab<1$ was previously open apart from numerical evidence.

September 2, 2026 claimed resolution

A preprint, “The frame set of the first Hermite function,” claims that a Wronskian construction together with a Gaussian zero-density theorem forces every subcritical non-frame lattice product to equal $(q-1)/q$ , completing the conjectured characterization. The claim is presented in an unrefereed preprint.

Current status (as of September 2026): A preprint claims the first-Hermite-function conjecture is solved, but its proof remains unverified; absent that verification, the result is not established.

Sources

No solutions have been posted yet.