Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

@article{MartnezFinkelshtein2009CriticalMQ,
  title={Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials},
  author={Andrei Mart{\'i}nez-Finkelshtein and Evguenii Rakhmanov},
  journal={Communications in Mathematical Physics},
  year={2009},
  volume={302},
  pages={53-111}
}
We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete… 

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