On mother body measures with algebraic Cauchy transform

Abstract

Abstract. Below we discuss the existence of a motherbody measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in C. Several conjectures and open problems are posed.

A refinement for rational functions of Pólya's method to construct Voronoi diagrams

“…We emphasize the fact, contained in the above theorem, that the logarithmic potential of the asymptotic measure is the global maximum of a fixed finite number of harmonic functions, and not only a local maximum (as in e.g. [2,3,5]). In addition, (Section 5.1), the Cauchy transform of the asymptotic measure trivially satisfies an algebraic equation (Corollary 1).…”

“…An algebraic equation for the Cauchy transform of an asymptotic measure is an interesting invariant, from which the local behaviour of the logarithmic potential may be deduced, see e.g [5]. Here the solutions of the algebraic equation have no monodromy, since they are rational functions, making the situation very simple.…”

Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

“…C + ν (t) − C − ν (t) dt ∈ R * , t ∈ γ, and then we get −Q (t) dt 2 > 0, t ∈ γ, which shows that γ is a horizontal trajectory of the quadratic differential −Q (z) dz 2 on the Riemann sphere. For more details, we refer the reader to [28], [30], [32], [27]. In the end, this analysis can be done in the case where p is a real polynomial without real zeros: Suppose that p (z) = (z − a) (z − b) (z − a) z − b with a = b ∈ C, (a) , (a) > 0.…”

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

“…Section 2 contains the proof of Theorem 1 and related results. The material presented in Section 4 is mostly borrowed from a recent paper [12] of the first author. It contains some general results connecting signed measures, whose Cauchy transforms satisfy quadratic equations, and related quadratic differentials in C. In particular, these results imply Theorem 2 as a special case.…”