Minimal Energy Point Systems on the Unit Circle and the Real Line

Gaál, Marcell Gábor and Nagy, Béla and Nagy-Csiha, Zsuzsanna and Révész, Szilárd (2020) Minimal Energy Point Systems on the Unit Circle and the Real Line. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 52 (6). pp. 6281-6296. ISSN 0036-1410

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Abstract

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of n electrons and n - 1 pairs of external protons of charge +1/2. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek; namely, we prove that any configuration of n electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by n - 1 pairs of protons.

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