On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy

Details On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy

2000-12-22

arxiv.org/abs/math-ph/0012040v1

Physics

Mathematical Physics

11 pages

Scientific paper

10.1088/0305-4470/34/16/318

A generalisation of the Stieltjes relations for the Painlev\'e-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher Painlev\'e-IV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schr\"odinger equations in the complex domain. As a corollary a new class of the Schr\"odinger operators with trivial monodromy is constructed in terms of the Painlev\'e-IV transcendents.

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