Potential-splitting approach applied to the Temkin–Poet model for electron scattering off the hydrogen atom and the helium ion

Abstract

The study of scattering processes in few body systems is a difficult problem especially if long range interactions are involved. In order to solve such problems, we develop here a potential-splitting approach for three-body systems. This approach is based on splitting the reaction potential into a finite range core part and a long range tail part. The solution to the Schrödinger equation for the long range tail Hamiltonian is found analytically, and used as an incoming wave in the three body scattering problem. This reformulation of the scattering problem makes it suitable for treatment by the exterior complex scaling technique in the sense that the problem after the complex dilation is reduced to a boundary value problem with zero boundary conditions. We illustrate the method with calculations on the electron scattering off the hydrogen atom and the positive helium ion in the frame of the Temkin-Poet model.

Potential splitting approach to e–H and e–He+scattering

“…It is worth noting that results for both the hydrogen scattering excluding the asymptotic Coulomb interaction and the He + scattering including this interaction converge equally well with respect to the splitting radius R. This means that our splitting procedure completely takes into account the asymptotic Coulomb interaction. Poet model of the same systems [11]. One can see that the influence of the corrections is drastically smaller here and decreases very fast with the splitting radius R. The reason for this is that the terms Ψ R 1 and Φ 1 decrease exponentially with R for the Temkin-Poet model [11] while they behave as inverse powers (17) for the full scattering problem.…”

Scattering Problem and Resonances for Three-Body Coulomb Quantum Systems: Parallel Calculations

“…As in the truncation region y α ≥ R, the difference (and the approximation error) goes to zero when R goes to infinity. More arguments on this subject can be found in papers [6,7] but the conclusive proof (like one in the two body case [5]) still remains to be done.…”

“…They are hard to implement especially for the Coulomb interactions. So let us describe here in short the potential splitting approach [5][6][7] which allows solving the Coulomb scattering problem without explicit use of the asymptotic form of the wave function.…”

“…In recent papers [5][6][7] there has been developed a method accurately solving the Schrödinger equation for the Coulomb scattering problem using ECS. This approach is based on the sharp splitting of the Coulomb potential for constructing the distorted incident wave which is generated by the tail part of the Coulomb potential.…”

Solving the Faddeev-Merkuriev Equations in Total Orbital Momentum Representation via Spline Collocation and Tensor Product Preconditioning

“…Even though the basic mathematical model for such a broad spectrum of physical systems is the Schrödinger equation, the diversity of model interactions and particular physical states leads to a variety of employed computational methods [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Thus, our ability to perform direct model-free calculations for such wide range of systems is of utmost importance for many branches of physics.…”