For a selfadjoint Schrödinger operator on the half line with a real-valued, integrable, and compactly-supported potential, it is investigated whether the boundary parameter at the origin and the potential can uniquely be determined by the scattering matrix or by the absolute value of the Jost function known at positive energies, without having the bound-state information. It is proved that, except in one special case where the scattering matrix has no bound states and its value is +1 at zero energy, the determination by the scattering matrix is unique. In the special case, it is shown that there are exactly two distinct sets consisting of a potential and a boundary parameter yielding the same scattering matrix, and a characterization of the nonuniqueness is provided. A reconstruction from the scattering matrix is outlined yielding all the corresponding potentials and boundary parameters. The concept of "eligible resonances" is introduced, and such resonances correspond to real-energy resonances that can be converted into bound states via a Darboux transformation without changing the compact support of the potential. It is proved that the determination of the boundary parameter and the potential by the absolute value of the Jost function is unique up to the inclusion of eligible resonances. Several equivalent characterizations are provided to determine whether a resonance is eligible or ineligible. A reconstruction from the absolute value of the Jost function is given, yielding all the corresponding potentials and boundary parameters. The results obtained are illustrated with various explicit examples.Mathematics Subject Classification (2010): 34A55 34L25 34L40 47A40 81U05 81U40
A method is presented to obtain the change in the potential and in the relevant wavefunction of a linear system of ordinary differential equations containing a spectral parameter, when that linear system is perturbed and a finite number of discrete eigenvalues are added to or removed from the spectrum. Some explicit formulas are derived for those changes by introducing certain fundamental linear integral equations for the corresponding unperturbed and perturbed linear systems. This generalized method is applicable in a unified manner on a wide class of linear systems. This is in contrast to the standard method for a Darboux transformation, which is specific to the particular linear system on which it applies. A comparison is provided in some special cases between this generalized method and the standard method for the Darboux transformation. In particular, when a bound state is added to the discrete spectrum, some Darboux transformation formulas are presented for the full-line Schrödinger equation, where those formulas resemble the Darboux transformation formulas for the half-line Schrödinger equation. The theory presented is illustrated with some explicit examples.
A system of linear integral equations is presented, which is the analog of the system of Marchenko integral equations, to solve the inverse scattering problem for the linear system associated with the DNLS (derivative nonlinear Schrödinger) equations. The corresponding direct and inverse scattering problems are analyzed, and the recovery of the potentials and the Jost solutions from the solution to the Marchenko system is described. When the reflection coefficients are zero, some explicit solution formulas are provided for the potentials and the Jost solutions in terms of a pair of constant matrix triplets representing the bound-state information for any number of bound states and any multiplicities. In the reduced case, when the two potentials in the linear system are related to each other through complex conjugation, the corresponding reduced Marchenko integral equation is obtained. The solution to the DNLS equation is obtained from the solution to the reduced Marchenko integral equation. The theory presented is illustrated with some explicit examples.
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