Uniqueness Theorems in Inverse Spectral Theory for One-dimensional Schrödinger Operators

New unique characterization results for the potential V (x) in connection with Schrödinger operators on R and on the half-line [0,∞) 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 Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.