Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I'm working on an article about propagators from int. representations of Green`s 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


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.


share|improve this question
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
add comment

1 Answer

The Pöschl–Teller potential is one of the few exactly solvable models out there, but it's not one with particularly nice solutions. I remember looking at it briefly a long while back in relation to coherent states and supersymmetric quantum mechanics. Its energy eigenstates are a class of 2F1 hypergeometric functions (or generalized Legendre functions in a special case).

I don't know enough to directly help you, but maybe some references will be of use. If not, you can clarify what you want by editing your question.

So, you can find solutions to the homogeneous differential equation and some references in (for example) Supersymmetric partners of the trigonometric Poschl-Teller potentials.

As for calculating the Green's function, I found two papers that use different methods and that might be useful:
Summing the spectral representations of Poschl–Teller and Rosen–Morse fixed-energy amplitudes;
Classification of Solvable Feynman Path Integrals (this is just a talk, for more details, follow the references).

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.