Solution to the interior transmission problem using nodes on a subinterval as input data

It is well known that a spherically symmetric wave speed problem in a bounded spherical region may be reduced, by means of Liouville transform, to the Sturm-Liouville problem B(a;q) in [0,1]. We prove that a twin dense subset of the nodal set in a subinterval (⊂[0,1]) uniquely determines the potential q on the whole interval [0,1] and the boundary parameter a. The method employed is to convert the inverse nodal problem into the inverse spectral problem with partial information given on the potential on a subinterval.