Inverse Spectral Analysis : the Case of Discrete Spectrum

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrr odinger operator H = ? d 2 dx 2 + q determine the potential completely. Included are theorems for nite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a nite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.