On the Determination of the Hydrogen Atom Equation from Two Spectra

It is well known that the two spectra {λn} and {μn} uniquely determine the potential function q (r) in a Sturm-Liouville equation defined on the unit interval having the singularity type ( ) 2 1 2 r r + +   (where  is a integer) at the point zero. In this work, we give the solution of the inverse problem on two partially noncoincide spectra for the Sturm-Liouville equation with the peculiarity in zero. In particularly in this case we obtain Hochstadt's theorem [1] concerning the structure of the difference ( ) ( ) q r q r −  . AMS subject classifications: 34A55 • 34B24 • 34L05