On the deformation of linear Hamiltonian systems

TL;DR

This paper studies how eigenvalues of linear Hamiltonian systems depend on their coefficients, deriving deformation and PDE equations, with applications to special functions and connections to monodromy and Lax pairs.

Contribution

It introduces a deformation framework for eigenvalues of Hamiltonian systems, especially in the 2x2 case, and relates it to monodromy and Lax pair theories.

Findings

Derived deformation equations for eigenvalues.

Applied results to generalized confluent Heun and Chandrasekhar-Page equations.

Connected eigenvalue deformations to monodromy preserving deformations.

Abstract

For linear Hamiltonian $2n\times 2n$ systems $J y'(x) = (\lambda W(x)+H(x))y(x)$ we investigate the problem how the eigenvalues $\lambda$ depend on the entries of the coefficient matrix $H$. This question turns into a deformation equation for $H$ and a partial differential equation for the eigenvalues $\lambda$. We apply our results to various examples, including generalizations of the confluent Heun equation and the Chandrasekhar-Page angular equation. We are mainly concerned with the $2\times 2$ case, and in order to reduce the degrees of freedom in $H$ as much as possible, we will first convert such systems into a complementary triangular form, which is a canonical form with a minimum number of free parameters. Furthermore, we discuss relations to monodromy preserving deformations and to matrix Lax pairs.