On Stieltjes relations, Painlev\'e-IV hierarchy and complex monodromy

TL;DR

This paper generalizes Stieltjes relations for Painlevé-IV transcendents, linking them to monodromy conditions of Schrödinger equations, and constructs new Schrödinger operators with trivial monodromy based on these solutions.

Contribution

It introduces a generalized form of Stieltjes relations for higher Painlevé-IV equations and connects them to monodromy conditions, leading to new Schrödinger operators with trivial monodromy.

Findings

Rational functions satisfying the generalized relations solve higher Painlevé-IV equations

A new class of Schrödinger operators with trivial monodromy is constructed

The approach interprets Stieltjes relations as local monodromy conditions

Abstract

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.