An Inverse Problem for Sturm-Liouville Operators on the Half-line Having   Bessel-type Singularity in an Interior Point
  We study the inverse problem of recovering Sturm-Liouville operators on the half-line with a Bessel-type singularity inside the interval from the given Weyl function. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided, also necessary and sufficient conditions for the solvability of the inverse problem are obtained. 
Alexey Fedoseev
math.SP math.CA