Uniqueness theorems in inverse spectral theory for one-dimensional Schrodinger operators

New unique characterization results for the potential V(x) in connection with Schrodinger operators on R and on the half-line [0, infinity) are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrodinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrodinger operators with confining potentials on the entire real line.