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Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Let S be a positivity‐preserving symmetric linear operator acting on bounded functions. The nonlinear equation −1m=z+Sm with a parameter z in the complex upper half‐plane ℍ has a unique solution m with values in ℍ. We show that the z‐dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3.