Green Function of Quantum Particle Moving in Two-dimensional Annular Potential

In this work, we present a new result which concerns the obtainment of the Green function relative to the time-independent Schrodinger equation in two dimensional space. The system considered in this work is a particle that have an energy E and moves in an axisymmetrical potential. Precisely, we have assumed that the potential (V(r)), in which the particle moves, to be equal to zero inside an annular region (radius b) and to be equal a positive constant (V-0) 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 bounded states regime for which (E < V-0). We have used, to obtain the Green function, the continuity of the solution and of its derivative at (r = b) and (r = a): We have obtained the associate Green function and the discrete spectra of the Hamiltonian in the region (r < b).