Eigenvalue estimates and trace formulas for the matrix Hill's equation (Q1585989)

The author considers the matrix Hill equation \[ -Y''+ Q(x) Y=\lambda Y,\quad Y\in C^k,\quad \lambda\in \mathbb{C},\quad Q(x)= Q(x+ 1), \] and investigates the behaviour of the eigenvalue sequence \(\lambda_n\) for periodic and related boundary conditions as \(|\lambda_n|\to\infty\) and develops trace formulas for pairs of these operators. The paper begins with a review on asymptotic expansion for the solution to the equation. The general results provide expansion for a Floquet matrix \(\psi_1(\lambda)\) (representing the translation by 1 on the vector space of the solution). Next, the author analyzes the Floquet multiplier and the periodic eigenvalues of the Hill operator, which are the roots of the entire function \(\text{det}(I-\psi_1(\lambda))\). (In practice, it takes help of an auxiliary function \(\overline\psi(\lambda)\) instead of \(\psi_1(\lambda)\).) The last section of the paper is devoted to the development of more refined estimates and power sum formulas for restricted pairs of Hill operators.