Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials (original) (raw)

Scattering and inverse scattering for the 1‐D Schrödinger equation with energy‐dependent potentials

Journal of Mathematical Physics, 1991

The one-dimensional Schriidinger equation with a potential k2 V(X) proportional to energy is studied. This equation is equivalent to the wave equation with variable speed. When V(X) < 1, is bounded below, and satisfies two integrability conditions, the scattering matrix is obtained and its asymptotics for small and large energies are established. The inverse scattering problem of recovering V(X) when the scattering matrix is known is also solved. By proving that all the solutions of a key Riemann-Hilbert problem have the same asymptotics for large energy, it is shown that the potential obtained is unique.

Inverse scattering theory for one-dimensional Schrödinger operators with steplike finite-gap potentials

Journal d'Analyse Mathématique, 2008

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.

Asymptotics of the scattering coefficients for a generalized Schrödinger equation

Journal of Mathematical Physics, 1999

The generalized Schr odinger equation d 2 =dx 2 + F (k) = ikP (x) + Q(x)] is considered, where P and Q are integrable potentials with nite rst moments and F satis es certain conditions. The behavior of the scattering coe cients near zeros of F is analyzed. It is shown that in the so-called exceptional case, the values of the scattering coe cients at a zero of F may be a ected by P (x): The location of the k-values in the complex plane where the exceptional case can occur is studied. Some examples are provided to illustrate the theory.

Scattering theory for the Schrödinger equation with repulsive potential

Journal de Mathématiques Pures et Appliquées, 2005

We consider the scattering theory for the Schrödinger equation with −∆ − |x| α as a reference Hamiltonian, for 0 < α ≤ 2, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where −|x| α is replaced by a general second order polynomial.

Scattering theory for Schrödinger operators with Bessel-type potentials

Journal für die reine und angewandte Mathematik (Crelles Journal), 2012

We show that for the Schrödinger operators on the half-axis with Bessel-type potentials κ(κ+1)/x 2 , κ ∈ [− 1 2 , 1 2), there exists a meaningful direct and inverse scattering theory. Several new phenomena not observed in the "classical case" of Faddeev-Marchenko potentials arise here; in particular, for κ = 0 the scattering function S takes two different values on the positive and negative semi-axes and is thus discontinuous both at the origin and at infinity.

Scattering by one-dimensional smooth potentials: between WKB and Born approximation

The paper discusses the applicability of WKB and Born (small perturbations) approximations in the problem of the backscattering of quantum particles and classical waves by one-dimensional smooth potentials with small amplitudes compared to the energy of the incident particle (above-barrier scattering). Both deterministic and random potentials are considered. The dependence of the reflection coefficient and localization length on the amplitude and the longitudinal scale of the scattering potential is investigated. It is shown that perturbation and WKB theories are inconsistent in the above-barrier backscattering problem. Not only the solutions but the regions of validity of both methods as well depend strongly on the details of the potential profile, and are individual for each potential. For deterministic potentials, a simple criterion that allows determining the boundary between the applicability domains of WKB and Born approximations is found. r

Small-energy analysis for the selfadjoint matrix Schrödinger operator on the half line. II

Journal of Mathematical Physics, 2014

The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the small-energy asymptotics for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.

Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line

Recently A. G. Ramm (1999) has shown that a subset of phase shifts δ l , l = 0, 1,. . ., determines the potential if the indices of the known shifts satisfy the Müntz condition l =0,l∈L 1 l = ∞. We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.

Small-energy analysis for the selfadjoint matrix Schroedinger operator on the half line. II

2013

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a second moment, it is shown that the corresponding scattering matrix is differentiable at zero energy. An explicit formula is provided for the derivative of the scattering matrix at zero energy. The previously established results when the potential has only the first moment are improved when the second moment exists, by presenting the small-energy asymptotics for the related Jost matrix, its inverse, and various other quantities relevantc to the corresponding direct and inverse scattering problems.