One-dimensional Quantum Scattering for Potentials Deened as Measures

Marseille Cedex 9 France One-dimensional quantum scattering for potentials de ned as measures Charles-Antoine GUERIN1 Abstract We generalize the basic one-dimensional scattering formalism to potentials de ned as measures and retrieve the classical results that hold for smooth potentials. We introduce a set of generalized eigenfunctions for the corresponding Schr odinger operator and study their analytical properties. This allows a characterization of the spectrum and an eigenfunction expansion. We also prove the existence and completeness of the wave operators and give explicit formulae for these latter. Key-Words : potential scattering, measure potential, eigenfunction expansion, wave operators 1991 MSC: 28A, 45, 46F, 46G, 81U05 Number of gures: 0 October 1997 CPT-97/P. 3553 anonymous ftp or gopher: cpt.univ-mrs.fr