Inverse spectral analysis for singular differential operators with matrix coefficients

Let $L_alpha$ be the Bessel operator with matrix coefficients defined on $(0,infty)$ by $$ L_alpha U(t) = U''(t)+ {I/4-alpha^2over t^2}U(t), $$ where $alpha$ is a fixed diagonal matrix. The aim of this study, is to determine, on the positive half axis, a singular second-order differential operator of $L_alpha+Q$ kind and its various properties from only its spectral characteristics. Here $Q$ is a matrix-valued function. Under suitable circumstances, the solution is constructed by means of the spectral function, with the help of the Gelfund-Levitan process. The hypothesis on the spectral function are inspired on the results of some direct problems. Also the resolution of Fredholm's equations and properties of Fourier-Bessel transforms are used here.