Van Vleck spectra of high-order Heun operators:\ finite-band universality and exterior asymptotics

Abstract

We study high-order analogues of the classical Heun operator of Fuchs index one, \[ \dq=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}, \qquad \deg Q_i\le i+1, \qquad \deg Q_k=k+1. \] For a fixed degree $n$ we consider the linear Van Vleck polynomials $V$ for which $\dq+V$ has a polynomial solution of degree $n$, and we form the spectral polynomial $Sp_n$ whose zeros are the zeros of these Van Vleck polynomials. The main result is a finite-band determinant representation and the resulting universality theorem: after normalization, all fixed power sums of the zeros of $Sp_n$ have limits given by explicit constant-term formulae depending only on the leading coefficient $Q_k$. The lower coefficients of $\dq$ enter only lower order correction terms. Combining this with the localization theorem for Van Vleck roots, we strengthen the usual germ-at-infinity conclusion to locally uniform convergence of the normalized Cauchy transforms and logarithmic potentials on the whole exterior of the convex hull of the zeros of $Q_k$. We also prove a determinacy criterion: if the spectral roots are asymptotically confined to a compact set with empty interior and connected complement, then the finite-band moments determine the actual weak limit. In particular, when the zeros of $Q_k$ are collinear the root-counting measures of $Sp_n$ converge weakly to a probability measure supported on the corresponding segment; this limit is independent of all lower coefficients of $\dq$. Finally, we prove holonomicity of the exterior Cauchy transform and derive Picard--Fuchs equations for the WKB periods, with an explicit third-order equation in the first non-classical case $k=3$. The paper ends with a precise mother-body conjecture for the genuinely complex case, clearly separated from the unconditional results.