Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials (Q627506)

The authors study the classical problem of the \(n\)-large asymptotic distribution of zeros of the Heine-Stieltjes polynomials. The latter are polynomial solutions to the second order linear ODEs with polynomial coefficients, \(A(z)y''+B(z)y'-n(n+\alpha-1)V_n(z)y=0\). Here, \(A(z)=\prod_{i=0}^p(z-a_i)\), \(B(z)=A(z)\sum_{i=0}^p\frac{\rho_i}{z-a_i}=\alpha z^p+\dots\), and \(V_n(z)\) is a monic polynomial of degree \(p-1\) called the van Vleck polynomial. The study is based on the fact that the zeros of an \(n\)-th degree Heine-Stieltjes polynomial correspond to (unstable) equilibrium configurations of \(n\) charges in the external field generated by \(p+1\) complex charges fixed at the given complex points \(a_i\). The authors prove that the sequence of the discrete critical measures related to the zero distribution has a weak \(n\)-large limit which is a continuous critical measure. The latter is defined by the authors as the critical point of the logarithmic energy functional with respect to certain variations of the measure the functional depends on. Thus the study of the limiting zero distribution of the Heine-Stieltjes polynomials reduces to the study of the continuous critical measure. The authors prove that the support of any such measure consists of a union of the level lines \(\mathop{Re}\int^z\sqrt{R(t)}\,dt=\text{constant}\), where \(R(z)\) is rational map. Moreover, all such level lines are either closed or they join two finite critical points. Furthermore, assuming that there is the \(n\)-large limit \(V(z)\) of the sequence of the Van Vleck polynomials \(V_n(z)\), the authors prove that \(R(z)=V(z)/A(z)\). This is used in a detailed study of the simplest nontrivial case \(p=2\) when \(V(z)=z-v\), \(v=\text{constant}\). In particular, the authors describe the set of all admissible values of \(v\). Some of the authors' findings are extended to the case of general \(p\).