Isoharmonic deformations and constrained Schlesinger systems

TL;DR

This paper introduces isoharmonic deformations related to Chebyshev polynomials on multiple intervals, deriving constrained Schlesinger equations and explicit solutions using hyperelliptic curves, with applications in integrable systems.

Contribution

It generalizes Chebyshev dynamics to isoharmonic deformations and formulates constrained Schlesinger equations with explicit hyperelliptic solutions, advancing algebraic geometry and integrable systems.

Findings

Derived explicit solutions to constrained Schlesinger equations.

Formulated and solved constrained Jacobi inversion for hyperelliptic curves.

Connected the theory to applications in billiards within ellipsoids.

Abstract

We introduce and study the dynamics of Chebyshev polynomials on $d>2$ real intervals. We define isoharmonic deformations as a natural generalization of the Chebyshev dynamics. This dynamics is associated with a novel class of constrained isomonodromic deformations for which we derive the constrained Schlesinger equations. We provide explicit solutions to these equations in terms of differentials on an appropriate family of hyperelliptic curves of any genus $g=d-1\ge 2$. The verification of the obtained solutions relies on the combinatorial properties of the Bell polynomials and on the analysis on the Hurwitz spaces. From the point of view of the classical algebraic geometry we formulate and solve the problem of constrained Jacobi inversion for hyperelliptic curves. We discuss applications of the obtained results in integrable systems, e.g. billiards within ellipsoids in $\mathbb R^d$.