Shapiro–Tater conjecture on root-counting measures of Heun spectral polynomials

Associate to $Sp_n(\lambda)$ the finite measure

$$\mu_n=\frac{1}{n+1}\sum_{j=1}^{n+1}{\delta(z-t_{n,j})},$$

where $\delta(z-a)$ is the Dirac measure supported at $a$. The measure $\mu_n$ is the root-counting measure of the spectral polynomial $Sp_n(\lambda)$. For an equation of the form

$$Q(z)\frac{d^2S}{dz^2}+P(z)\frac{dS}{dz}+V(z)S=0,$$

let $Conv_Q$ denote the relevant convex hull of the roots of $Q(z)$, and let $\{\mu_n\}$ be the sequence of root-counting measures of its spectral polynomials. Shapiro–Tater conjecture. For any equation of the stated form, the sequence $\{\mu_n\}$ converges to a probability measure $\mu$ supported on the union of three curved segments inside $Conv_Q$, each connecting one of the three roots of $Q(z)$ to a certain interior point. Moreover, the limiting measure $\mu$ depends only on $Q(z)$ and is independent of $P(z)$. Extensive numerical experiments motivate this conjecture; the asserted convergence, the described support, and independence from $P(z)$ are not established in the supplied text.