Higher-order differential equations having a singularity in an interior point (Q1411439)

This paper deals with the differential equation \[ y^{(n)}(x)+\sum^{n-2}_{j=0}\left(\frac{\nu_{j}}{(x-a)^{n-j}}+q_{j}(x)\right)y^{(j)}(x)=\lambda y(x),\qquad 0<x<T, \] which possesses a nonintegrable singularity at an interior point \(a\in (0,T)\). Additional matching conditions are imposed at the singular point \(x=a\) via a transition matrix which connect solutions of the equation near the singular point. The matching conditions involve special fundamental systems of solutions to the equation. Asymptotical, analytical and structural properties of these systems and the behavior of the corresponding Stokes multipliers are studied in detail. Using these properties, the author obtains asymptotical formulae for the solutions which satisfy the matching conditions. Various asymptotic estimates on the eigenvalues of boundary value problems for the equation are also given. Furthermore, he introduces the Weyl matrix for the equation and discusses the inverse problem of recovering the equation from the given Weyl matrix. He proves that the Weyl matrix determines the equation uniquely.