# Propagators from integral representations of Greens functions

I'm working on an article about propagators from int. representations of Greens functions for several N-dimensional potential(all this is done in an N-dimensional Euclidian space). Potentials like the free-particle, harmonic oscillator, Coulomb and Poschl-Teller. I started from the radial equation which is satisfied by the Qth partial-wave Green's function

$$\biggl[E+\frac{1}{2r^{N-1}}\frac{\partial}{\partial r}r^{N-1}\frac{\partial}{\partial r}-\frac{Q(Q+N-2)}{2r^{2}}-V(r)\biggr]G_{Q}^{N}[r,r',E]=\frac{\delta(r-r')}{(rr')^{N/2-1/2}}$$

As the usual procedure goes, the construction of the Green's function is

$$G[r,r',E]=\frac{u(r_{-})v(r_{+})}{\frac{1}{2}r^{N-1}W[u,v]}$$

obviously, here the u and v are solutions of the homogeneous equation with the appropiate boundary conditions and W is the Wronskian. With the first three potential I had no problem to arive at the N-dimensional form of the propagator.

But I don't know what method to use for the Poschl-Teller potential, I can't even find the solutions for the homogeneous equation with this given potential. Do you have any pointers on this problem?I tried the classical method used to solve the Schrodingers eq. with this potential but I got nowhere with it.

Thanks.

• I edited your math a bit to make it look cleaner; hopefully I didn't introduce any inaccuracies. (Also, great question, though unfortunately I don't know enough to answer it without doing a fair amount of reading first...) – David Z Sep 20 '11 at 2:38