Accessory parameters of the Heun equation

Definitions and mathematical statement

Fix distinct $a_0,a_1,a_2\in\mathbb C$, real numbers $\alpha_0,\alpha_1,\alpha_2>0$, and real $\alpha',\alpha''$ satisfying $\alpha_0+\alpha_1+\alpha_2+\alpha'+\alpha''=2$. Put $A=\alpha'\alpha''$ and consider the family \[ y''+\left(\sum_{j=0}^2\frac{1-\alpha_j}{z-a_j}\right)y' +\frac{Az-\lambda}{(z-a_0)(z-a_1)(z-a_2)}y=0, \qquad\lambda\in\mathbb C. \] It has regular singularities at the three finite marked points and at infinity. The variable $\lambda$ is the accessory parameter; all other data are held fixed.

For two independent local solutions, analytic continuation of their ratio around punctures gives a group of Möbius transformations, well-defined up to conjugacy. Let $E$ consist of the parameters for which this group is conjugate to a subgroup of $\operatorname{PSU}(2)$, the group induced by unitary $2$-by-$2$ matrices of determinant one on the Riemann sphere.

Describe $E$ explicitly in terms of the prescribed data. In particular, characterize exactly when it is nonempty and give an explicit upper bound for its cardinality in terms of the exponent data $\alpha_j,A$. Also answer these questions in the real version, where the marked points are real and only real $\lambda$ are admitted. The primary source records that $E$ is finite; the target is effective description, existence criteria, and quantitative bounds, rather than reproving finiteness.

Short English statement

Describe and count the Heun accessory parameters producing spherical unitary monodromy, including the real-parameter case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700002 (AMR-036-0002), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Heun equation problem (updated 2020). Original source indexed by AMR-036-0002. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Accessory parameter of the Heun equation. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/heun.pdf.