Inverse spectral analysis for singular differential operators with matrix coefficients

Abstract

Let Lα be the Bessel operator with matrix coefficients defined on (0, ∞) by LαU(t) = U″ (t) + I/4 - α2 / t2 U(t), where α 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 Lα + 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.