On the deformation of linear Hamiltonian systems

Harald Schmid

doi:10.1016/j.jmaa.2021.125051

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arXiv · 2010.05175

On the deformation of linear Hamiltonian systems

Harald Schmid

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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.

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Harald Schmid. 2020-10-11. On the deformation of linear Hamiltonian systems. https://doi.org/10.1016/j.jmaa.2021.125051

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