Abstract In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes large-distance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, and phase shifts are given without large-distance asymptotics. We give a discussion of one- and two-dimensional scatterings. Moreover, we also suggest a dimensional renormalization scheme to remove the divergence encountered in singular-potential scattering.
Abstract In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes large-distance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, and phase shifts are given without large-distance asymptotics. We give a discussion of one- and two-dimensional scatterings. Moreover, we also suggest a dimensional renormalization scheme to remove the divergence encountered in singular-potential scattering.
A phase-shift analysis of quantum mechanical scattering problems in one dimension is developed. Many of the techniques and results of three dimensions remain even in one dimension, while the discussion is vastly simplified, and thus accessible to the student much earlier, due to the absence of special functions (in particular, Bessel and Legendre functions) with their complicated asymptotic formulas and orthogonality relations. A “partial wave” analysis leads to the familiar relation fl∼eiδl sinδl between the scattering amplitude in the lth partial wave and the phase shift δl. The optical theorem is evident as an immediate consequence of the conservation of probability. The method is applied to an analysis of the well-known square-well scattering resonances.
Long-range interactions and, in particular, two-body potentials with power-law long-distance tails are ubiquitous in nature. For two bosons or fermions in one spatial dimension, the latter case being formally equivalent to three-dimensional s-wave scattering, we show how generic asymptotic interaction tails can be accounted for in the long-distance limit of scattering wave functions. This is made possible by introducing a generalization of the collisional phase shifts to include space dependence. We show that this distance dependence is universal, in that it does not depend on short-distance details of the interaction. The energy dependence is also universal, and is fully determined by the asymptotic tails of the two-body potential. As an important application of our findings, we describe how to eliminate finite-size effects with long-range potentials in the calculation of scattering phase shifts from exact diagonalization. We show that even with moderately small system sizes it is possible to accurately extract phase shifts that would otherwise be plagued with finite-size errors. We also consider multichannel scattering, focusing on the estimation of open channel asymptotic interaction strengths via finite-size analysis.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
We give a full and self contained account of the basic results in N-body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r-μ, [Formula: see text]. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t → ±∞ for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions.
The normalization problem of a wave function for an arbitrary one-dimensional field is considered. It is shown, that for an infinite motion type the magnitude of the normalization constant does not depend on the form of a scattering potential and it is determined by the boundary conditions imposed on the character of the investigated motion. The general formula for the normalization constant as a function of amplitudes of the waves converging to the potential is derived. Keyword: Normalization problem, wave function, one-dimensional motion, arbitrary potential PACS: 03.65.Nk, 02.30.Em
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.