Spectral Asymptotics for the Systems of Differential Equations and Applications

We shall present in the talk new results on asymptotic representations for the fundamental system of solutions of the equation
$$ y' +Q(x)y = \lambda A(x) y, \quad x\in [0,1], $$
with respect to $\lambda \to\infty$. Here $Q$ and $A$ are $n\times n$ matrices. In particular, we present explicit formulae for the first $k$ terms of the asymptotic expansions under minimal assumptions on the smoothness of the matrices $Q$ and $A$. As application we get results on the unconditional convergence of the Fourier series in the eigenfunctions of the operators generated by the systems of differential equations and by the scalar differential high order operators.
The talk is based on the joint work with A. P. Kosarev.