Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An oval is a smooth closed strictly convex plane curve. Its outer billiard reflects an exterior point in the midpoint of a tangent segment, with a consistent tangent branch. A periodic outer orbit determines a circumscribed polygon of extremal area. The area spectrum is the set of these polygonal areas. The question is a spectral research programme and does not specify an operator in advance.

Statement recorded in the supplied source.

For an oval $\gamma$, the area spectrum of its outer billiard is the set of areas of the circumscribed polygons formed by periodic outer-billiard trajectories. Is this area spectrum related to the spectrum of a differential operator? [S2]

Short English statement

Find whether the areas of periodic outer-billiard polygons can be related to the spectrum of a differential operator.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200017 (AMR-051-0017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Serge Tabachnikov, Problem 5, pp. 12--13. https://arxiv.org/pdf/2110.10750v2.