Integral equations of scattering in one dimension

Variational Approximations in a Path-Integral Description of Potential Scattering

Using a recent path integral representation for the T-matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables -- over which one has to integrate functionally -- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading).We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to th…

A novel way of calculating scattering integrals

The technique coined as NDIM - Negative Dimensional Integration Method by their discoverers, relies on a three-pronged basis: Gaussian integration, series expansion and analytic continuation. The technique has been successfully applied to the calculation of covariant and non covariant Feynman integrals in a generic dimensional regularization space, i.e., D-dimensional space-time for D including the negative domain values. Since the dimensionality is general, we can use specifically for one-dimensional integrals. In this work we show how this technique can be applied to tackle certain improper …

Feynman Integral in Quantum Walk, Barrier-top Scattering and Hadamard Walk

The aim of this article is to relate the discrete quantum walk on $\mathbb{Z}$ with the continuous Schrödinger operator on $\mathbb{R}$ in the scattering problem. Each point of $\mathbb{Z}$ is associated with a barrier of the potential, and the coin operator of the quantum walk is determined by the transfer matrix between bases of WKB solutions on the classically allowed regions of both sides of the barrier. This correspondence enables us to represent each entry of the scattering matrix of the Schrödinger operator as a countable sum of probability amplitudes associated with the paths of the qu…

Variational Methods for Path Integral Scattering

In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential.…

Legendre函数在Rosen-Morse散射态中的推广

Generalization of Legendre functions applied to Rosen-Morse scattering states

本文推广连带Legendre函数，以超几何函数显式描述Rosen-Morse势的散射态，给出反射/透射系数及谱测度，提供无需路径积分的完整经典解。

The three-body Coulomb scattering problem in discrete Hilbert-space basis representation

For solving the $2\to 2,3$ three-body Coulomb scattering problem the Faddeev-Merkuriev integral equations in discrete Hilbert-space basis representation are considered. It is shown that as far as scattering amplitudes are considered the error caused by truncating the basis can be made arbitrarily small. By this truncation also the Coulomb Green's operator is confined onto the two-body sector of the three-body configuration space and in leading order can be constructed with the help of convolution integrals of two-body Green's operators. For performing the convolution integral an integration co…