Uniqueness of the interior transmission problem with partial information on the potential and eigenvalues

The inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b. A uniqueness theorem for the potential q   of the derived Sturm–Liouville problem B(q)B(q) is presented from the data involving fractions of the eigenvalues of the problem B(q)B(q) on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl-function techniques and properties of zeros of a class of entire functions.