Solving the Coulomb scattering problem using the complex-scaling method

Abstract

Abstract. -Based on the work of Nuttall and Cohen [Phys. Rev. 188 (1969) 1542] and Resigno et al. [Phys. Rev. A 55 (1997) 4253] we present a rigorous formalism for solving the scattering problem for long-range interactions without using exact asymptotic boundary conditions. The long-range interaction may contain both Coulomb and short-range potentials. The exterior complex scaling method, applied to a specially constructed inhomogeneous Schrödinger equation, transforms the scattering problem into a boundary problem with zero boundary conditions. The local and integral representations for the scattering amplitudes have been derived. The formalism is illustrated with numerical examples.Introduction. -Few-body systems held together by a mutual Coulomb interaction are of great interest in many areas of quantum physics. However, solving the Coulomb scattering problem is a very difficult task both from the theoretical as well as the computational points of view due to the long-range character of the Coulomb interaction. The asymptotic boundary conditions for the wave function at large separations between particles are already complicated for the few-body scattering problem with shortrange interactions [1]. They become even more complicated for the long-range case when the Coulomb potential is present in the interaction [2]. Therefore, a method which allows the problem to be solved without explicit use of the asymptotic form of the wave function is of great interest from both the theoretical and computational points of view.One of such methods was proposed by Nuttal and Cohen [3]. The approach is based on the complex scaling theory [4]. The idea can briefly be formulated as follows.The Schrödinger equation is recast into its inhomogeneous (driven) form by splitting the wave function into the sum Ψ = Ψ in + Ψ sc of the incident Ψ in and scattered Ψ sc waves as

Quantum Scattering with the Driven Schrödinger Approach and Complex Scaling

“…Another useful observation made from the representation (8) is that the scattering amplitude A R is completely determined by that part of the solution R , which is restricted on the finite domain 0 ≤ r ≤ R. After the ECS transformation of the coordinate, the boundary condition (5) becomes the zero boundary condition, and the Schrödinger equation (2) can be easily solved. Summarizing the description of our approach, we can say that the only parameter affecting the results is the radius R. It is important that the scattering problem (1) is exactly reduced to the boundary value problem on the interval [0, R] for an arbitrary finite value R [10]. For the numerical calculations, however, we need to use the asymptotics of the wave function at the right boundary R. This gives an error, those magnitude is defined by the accuracy of the asymptotics.…”

“…Using (6) with δ R = η log 2k R − σ , we calculate the phase shift δ R and finally reconstruct δ with the relation δ = δ R + δ R . The numerical implementation of this approach has recently been shown to have both good accuracy and high efficiency [10].…”

“…Here, the third particle momentum q (10) into Schrödinger equation (9), we get its inhomogeneous (driven) form…”

“…3 we give an outline of a new proof showing that the method of [9], in the two-body case, can be rigorously generalized to include a potential composed of a short range potential and a long range Coulomb contribution. For a more complete discussion of this work we refer to recent work of Volkov et al [10]. We then outline how to generalize the method [10] to a many channel, three-body problem in Sect.…”

“…For a more complete discussion of this work we refer to recent work of Volkov et al [10]. We then outline how to generalize the method [10] to a many channel, three-body problem in Sect. 4.…”