Energy Dependent Inverse Scattering

Abstract

Abstract. The paper discusses the inverse scattering problem to recover the potentials of an energy dependent Schrö dinger equation from the scattering data. A necessary and su‰cient condition for a given function to be the scattering data is obtained. The potential can be recovered by a simple formula from the solution of a Marchenko type equation with a function determined from the scattering data.

An inverse problem for the quadratic pencil of non-self-adjoint matrix operators on the half-line

“…According to (13), Φ * (x, ρ), Φ(x, ρ) does not depend on x. Using (8), (12), (14) and the asymptotics (i 4 ) of Theorem 1, we get…”

“…Inverse spectral problems consist in recovering operators from their spectral characteristics. In the scalar case, inverse problems for quadratic pencils were studied in the works [7][8][9] (half-line), [10][11][12] (full line) and [13][14][15][16][17] (finite interval).…”

Hill's operators with the potentials analytically dependent on energy

“…Schrödinger operators with polynomially energy-dependent potentials are also well studied, see, e.g., Alonso [A80], Jaulent and Jean [JJ76], [JJ76x], Kamimura [Ka08], see also the book [Ma12] and references therein, moreover, there is enormous physical and technical literature on this subject. By the well-known technique developed by Keldysh [Ke71], these problems are reduced to vector spectral problems where the potential does not depend on the spectral parameter.…”

Inverse Scattering on the Half-Line for ZS-AKNS Systems with Integrable Potentials

“…Energy-dependent Schrödinger equations are also used to describe vibrations of mechanical systems in viscous media [82]. In [37][38][39][40], the authors studied the inverse scattering problems for energy-dependent Schrödinger equations with regular potentials; see also the papers [4,43,59,66,67,74,78,79]. It turns out that the scattering problem for (1.5) can be transformed to the one for the ZS-AKNS system of the form (1.1)-(1.2).…”