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The Scattering Operator and the Adiabatic Theorem
The Inverse Scattering Problem I
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adiabatic theorem amplitude analytic continuation assume bound boundary conditions commutes with H completeness relation consider continuous spectrum corresponding defined denote derivative diagonal dielectric differential equation discrete spectrum eigenfunctions eigenfunctions of H eigenstates eipx electromagnetic expression extended operator follows function of q function V(x Gelfand-Levitan equation given H. E. Moses Hamiltonian Hence Hermitian Hilbert space integral equation inverse problem kernel Lemma Levitan Mathematical Sciences Nuovo Cimento obtain one-dimensional operator H paper perturbed point eigenvalues poles positive definite potential V(x prescribed procedure proof quadratically integrable rational function reflection coefficient reflection coefficient b(k representation result satisfy scattering matrix scattering operator scattering potential scattering problem Schrodinger equation shown sinh solve spectral measure function spectrum of H substituting sufficient condition theory tion transformation transmission coefficient triangularity conditions unperturbed Hamiltonian upper half-plane values vanishes variables vector space weight operator York University