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

New unique characterization results for the potential V ( x ) V(x) in connection with Schrödinger operators on R R and on the half-line [ 0 , ∞ ) [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.