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 axi-symmetrical 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 ($r<b$).