Explicit Quantum Green's Functions on a Piecewise Continuous Symmetrical Spherical Potential

This work provides new results which are related to the calculation of Green's function for time-independent Schrödinger equation in three-dimensional space. Particularly, we have focused on computing the quantum Green's function for two kinds of spherical potentials. In the first case, we assumed that the potential is a piecewise continuous potential V(r) which means to be equal to a constant V0 inside the sphere (radius a) and it is equal to zero outside the sphere. For this potential, to derive the Green function, we have used continuity of the solution and discontinuity of its first derivative at r = a (at the edge). We have derived the solution in two cases: E > V0 and E < V0. For the second kind of potential, we have assumed that the potential V(r) is equal to a negative constant -V0 inside the sphere, and it is equal to zero outside it. Also, we used the continuity of its solution and discontinuity of its derivative at the edge (r = a) to obtain the associated Green's function which shows the discrete spectra of the Hamiltonian when - V0 < E < 0.