Reconstruction formula for differential systems with a singularity

Our studies concern some aspects of scattering theory of the singular differential systems $y'-x^{-1}Ay-q(x)y=\rho By$, $x>0$ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl-type solutions of the considered differential system. Formulas of such a type play an important role in constructive solution of inverse scattering problems: use of such formulas, where the terms in their right-hand sides are previously found from the so-called main equation, provides a final step of the solution procedure. In order to obtain the above-mentioned reconstruction formula, we establish first the asymptotical expansions for the Weyl-type solutions as $\rho\to\infty$ with $o\left(\rho^{-1}\right)$ rate remainder estimate.