An inverse scattering problem for a perturbed Hill's operator

Trace Formulas for Schrödinger Operators in Connection with Scattering Theory for Finite-gap Backgrounds

“…Just differentiate (3.3) with respect to z, which is permissible by uniform convergence on compact subsets of C\{E j } 2g j=0 . We remark that if we require our perturbation to satisfy the usual short-range assumption as in [1], [9,10,11] (i.e., the first moment is integrable, see (5.1)), then we even have e ∓ixk(z) (ψ ± (z, x) −ψ q,± (z, x)) → 0.…”

“…However, most results are in connection with scattering theory for a constant background. On the other hand, scattering theory for one-dimensional Schrödinger operators with periodic background is a much older topic first investigated by Firsova in a series of papers [9]- [11]. Nevertheless, many questions which have long been answered in the constant background case are still open in this more general setting.…”

Perturbations of periodic Sturm--Liouville operators

“…The question whether eigenvalues accumulate at the band edges has a long tradition going back to the seminal results of Rofe-Beketov [10], which were later extended by Schmidt [11] (see also [1, §5.4] for the special case r 0 = r 1 = 1 and p 0 = p 1 ). They play also an important role for the scattering theory in this setting [2,3,4,7]. The currently best results in this direction can be found in [8], which apply in the special case r 0 = r 1 .…”

Characterization of the scattering data for the Sturm–Liouville operator

“…Being a natural extension of the classical case c 1 = c 2 = 0, the inverse scattering problems for Equation with ρ ( x ) = 1 have been investigated in . Also note that in more general form, but under the second moment condition, the inverse scattering problems for one‐dimensional Schrödinger operators with steplike finite gap potentials in various settings have been widely investigated in series of papers .…”