Research Problems in Function Theory — Problem 2.71

Definitions and mathematical statement

Let $H\in\mathbb C[z]$ be a nonconstant polynomial and consider the linear differential equation \[ y''(z)+H(z)y(z)=0\qquad(z\in\mathbb C). \] Its solutions are entire. Characterize those polynomials $H$ for which there exists a nonzero solution $y$ with infinitely many zeros, every one of which lies on the real axis. Multiplicities do not create an ambiguity here: a nontrivial solution of this equation cannot have both $y(z_0)=0$ and $y'(z_0)=0$. The condition is imposed on one solution. Requiring two linearly independent solutions with only finitely many nonreal zeros gives a much more restrictive situation, mentioned separately in the source. Likewise, a zero-free solution does not qualify, and allowing finitely many nonreal zeros changes the exact classification problem. One can express the requested class as \[ \mathcal P=\{H\in\mathbb C[z]\setminus\mathbb C: \exists y\not\equiv0, y''+Hy=0, \#y^{-1}(0)=\infty, y^{-1}(0)\subset\mathbb R\}. \] Seek necessary and sufficient conditions on the coefficients or associated differential equation for membership in $\mathcal P$. The source supplies linear and certain quartic examples, together with later partial structural results, but those examples are not a proposed exhaustive list.

Short English statement

Which polynomial differential equations $y''+Hy=0$ admit a solution with infinitely many zeros and no zeros off the real axis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302071 (AMR-022-2071), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.71. Original source indexed by AMR-022-2071. https://arxiv.org/abs/1809.07200.