Weighted-residual methods for the solution of two-particle Lippmann-Schwinger equation without partial-wave decomposition

Calculation of eigenvalues by Greens-functions and the Lippmann-Schwinger equation

We calculate eigenvalues of one-dimensional quantum-systems by the exact numerical solution of the Lippmann-Schwinger equation, analogous to the scattering problem. To illustrate our method, we treat elementary problems: the harmonic and quartic oscillator, a symmetric and a skew double-well potential, and potentials with finite and infinite depth. Furthermore, we show how our method can be used for eigenvalue-engineering.

Machine Learning Regularized Solution of the Lippmann-Schwinger Equation

Solution of the discretized Lippmann-Schwinger equation in the spatial frequency domain involves the inversion of a linear operator specified by the scattering potential. To regularize this inevitably ill-conditioned problem, we propose a machine learning approach: a recurrent neural network with long short-term memory (LSTM) and with the null space projection of the Lippmann-Schwinger kernel on the recurrence path. The learning method is trained using examples of typical scattering potentials and their corresponding scattered fields. We test the proposed method in two cases: electromagnetic s…

共焦抛物线台球中Lippmann-Schwinger方程的解

Solutions of the Lippmann-Schwinger equation for confocal parabolic billiards

本文解析和数值求解了共焦抛物线台球的Lippmann-Schwinger方程，研究了共振、隧穿及特定能量下的透明现象。

Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners

In this work the Lippmann-Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann-Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for sma…

Recurrent Localization Networks applied to the Lippmann-Schwinger Equation

The bulk of computational approaches for modeling physical systems in materials science derive from either analytical (i.e. physics based) or data-driven (i.e. machine-learning based) origins. In order to combine the strengths of these two approaches, we advance a novel machine learning approach for solving equations of the generalized Lippmann-Schwinger (L-S) type. In this paradigm, a given problem is converted into an equivalent L-S equation and solved as an optimization problem, where the optimization procedure is calibrated to the problem at hand. As part of a learning-based loop unrolling…

Direct numerical solution of the two-particle Lippmann-Schwinger equation in coordinate space using the multi-variable Nystrom method

Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann Schwinger (LS) equation is considered as a means of avoiding the shortcomings of partial-wave expansion at high energies and in the context of few-body problems. Upon the regularization of the singular kernel of the three-dimensional LS equation by a subtraction technique, a three-variate quadrature rule is used to solve the resulting nonsingular integral equation. To avoid the computational burden of discretizing three variables, advantage is taken of the fact that, for central potentials,…