Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 16, pp. 1-19.

Title: Inverse spectral analysis for singular
       differential operators with matrix coefficients
       
Authors: Nour el Houda Mahmoud (Faculte des Sciences de Tunis, Tunisia)
         Imen Yaich (Faculte des Sciences de Tunis, Tunisia)

Abstract: 
 Let $L_\alpha$ be the Bessel operator with
 matrix coefficients defined on $(0,\infty)$ by
$$
 L_\alpha U(t) = U''(t)+  {I/4-\alpha^2\over t^2}U(t),
$$
 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.
 
Submitted October 14, 2005. Published February 2, 2006.
Math Subject Classifications: 45Q05, 45B05, 45F15, 34A55, 35P99.
Key Words: Inverse problem; Fourier-Bessel transform; 
    spectral measure; Hilbert-Schmidt operator; 
    Fredholm's equation.