Explicit Quantum Green Function for Scattering Problems in 2-D Potential

TL;DR

This paper derives an explicit Green function for a 2D quantum scattering problem with a specific piecewise potential, revealing resonance energies where reflection is absent, advancing analytical solutions in quantum scattering theory.

Contribution

It provides a new explicit Green function for a 2D Schrödinger equation with a piecewise axi-symmetrical potential, including resonance energy analysis.

Findings

Derived Green function for the specified potential

Identified resonance energies with no reflected waves

Enhanced analytical understanding of 2D quantum scattering

Abstract

In this work, we present a new result which concerns the derivation of the Green function relative to the time-independent Schrodinger equation in two dimensional space. The system considered in this work is a quantum particle that have an energy E and moves in an axi-symmetrical potential. Precisely, we have assumed that the potential V(r), in which the quantum particle moves, to be equal to zero inside a disk (radius b) and to be equal a positive constant V0 in a crown of internal radius b and external radius a (b < a) and equal zero outside the crown (r > a). We have explored the diffusion states regime for which E > V0. We have used, to obtain the Green function, the continuity of the solution and of its first derivative at r = b and r = a. We have obtained the associate Green function showing the resonance energies (absence of the reflected waves) for the case E > V0.