Inverse problems for radial Schrodinger operators with the missing part of eigenvalues

We study inverse spectral problems for radial Schrodinger operators in L-2(0, 1). It is well known that for a radial Schrodinger operator, two spectra for the different boundary conditions can uniquely determine the potential. However, if the spectra corresponding to the radial Schrodinger operators with the two potential functions miss a finite number of eigenvalues, what is the relationship between the two potential functions? Inspired by Hochstadt (1973)'s work, which handled the Sturm-Liouville operator with the potential q is an element of L-1(0, 1), we give a corresponding result for radial Schrodinger operators with a larger class of potentials than L-1(0, 1). When q is an element of L-1(0, 1), we also consider the case where the spectra corresponding to the radial Schrodinger operators with the two potential functions miss an infinite number of eigenvalues and the eigenvalues are close in some sense.