From Lippmann-Schwinger formulations to a general formula for absolute asymptotic scattering phase functions and shifts: a unified framework for potentials of any range
Appropriately handling the Lippmann-Schwinger equation can bring substantial benefits in quantum scattering. Here, the advantages of a parametric equation for the system's normalised wavefunction, ψ, supplied with regular fref and irregular gref wavefunctions of a reference Hamiltonian for arbitrary Vref=Ueff−V subsystems are discussed to extend previous results [H. Piel and M. Chrysos, Mol. Phys., 116, 2364 (2018)] to any interaction-range for the effective Ueff and ‘excess’ V potentials. While the idea of using reference wavefunctions to solve scattering problems is not new [J. P. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998); B. P. Ruzic, C. H. Greene, and J. L. Bohn, Phys. Rev. A 87, 032706 (2013)], convincing evidence that a wide variety of problems can benefit from the careful choice of fref and gref and the demonstration of how their occurrence in a compact formula permits the absolute definition of asymptotic scattering phases are among our highlights. Various examples employing Coulomb-van-der-Waals interactions in helium, along with an insightful rephrasing of ideas originally introduced in quantum defect theory for ultracold alkali-alkali collisions are given, adding to the recognition of these works and further promoting the need for properly selected reference wavefunctions in scattering calculations.
Appropriately handling the Lippmann-Schwinger equation can bring substantial benefits in quantum scattering. Here, the advantages of a parametric equation for the system's normalised wavefunction, ψ, supplied with regular fref and irregular gref wavefunctions of a reference Hamiltonian for arbitrary Vref=Ueff−V subsystems are discussed to extend previous results [H. Piel and M. Chrysos, Mol. Phys., 116, 2364 (2018)] to any interaction-range for the effective Ueff and ‘excess’ V potentials. While the idea of using reference wavefunctions to solve scattering problems is not new [J. P. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998); B. P. Ruzic, C. H. Greene, and J. L. Bohn, Phys. Rev. A 87, 032706 (2013)], convincing evidence that a wide variety of problems can benefit from the careful choice of fref and gref and the demonstration of how their occurrence in a compact formula permits the absolute definition of asymptotic scattering phases are among our highlights. Various examples employing Coulomb-van-der-Waals interactions in helium, along with an insightful rephrasing of ideas originally introduced in quantum defect theory for ultracold alkali-alkali collisions are given, adding to the recognition of these works and further promoting the need for properly selected reference wavefunctions in scattering calculations.
The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (1), and studied by Calogero in his book5, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines δℓ(k) as a continuous function of k for all k ≥ 0, up to infinity, where δℓ(∞) = 0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g> 0, m ≥ 2, etc. We also give an example of application to the low-energy behaviour of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.
A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
Quantum scattering connects microscopic dynamics to amplitudes, phase shifts, cross sections, transport coefficients, and resonances. This selective Review organizes the Lippmann–Schwinger equation, resolvents, transfer and scattering matrices, nonequilibrium Green’s functions, and R-matrix theory as related representations of a common scattering problem. Born, Wentzel–Kramers–Brillouin, and eikonal error tests accompany comparisons of numerical scaling, stability, and method selection. Inverse scattering is examined through the effects of phase loss, incomplete aperture, Jacobian null spaces, singular-value decay, model discrepancy, and noise, together with regularized, Bayesian, and learned approaches for stabilizing reconstruction. The Review then discusses probability-level optimal transport and its extension to emerging amplitude-based formulations, using the Quantum Optimal Transport Amplitude Method (QOTAM) as a concrete example. Its Euclidean and real-time variational constructions are summarized and tested across resonant barriers, multichannel and angular scattering, three-dimensional targets, and disordered waveguides. Independent reference calculations verify complex-amplitude closure, unitarity, flux conservation, and optical-theorem relations to benchmark-dependent numerical precision. In inverse problems, QOTAM is found to be competitive with correctly specified phase-sensitive likelihood methods while showing useful robustness in selected model-mismatch tests. These examples illustrate the promise of optimal-transport-based scattering methods. More broadly, the Review emphasizes how physical constraints, uncertainty quantification, identifiability, and computational scalability jointly shape reliable forward and inverse-scattering analyses.
Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length a of the He-He interaction and a three-body parameter a∗ as input, where a∗ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to 1/a∗ versus 1/a should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational 1/a∗ versus 1/a to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum J>0 are also discussed.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.