On the deformation of linear Hamiltonian systems

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.