# equation for the potential in terms of the phase shift for a rotational invariant potential

let be a potential in 3d invariant under rotations so $V(r)$

in the WKB approximation the phase shifts are given by

$$\delta _{l} (k)= \int_{a}^{\infty}dr \sqrt{k^{2}-2mV(r)-l(l+1)r^{-2}}-\int_{b}^{\infty}dr \sqrt{k^{2}-l(l+1)r^{-2}}$$

here 'a' and 'b' are the zeros of the integrands

http://books.google.es/books?id=Wq_u0r7fOaEC&pg=PA27&lpg=PA27&dq=phase+shift+of+scattering+in+the+WKB+approximation&source=bl&ots=0GUNjxZYlA&sig=ZTh1bC2jQ-o2NWSlg1Ld0WTLBME&hl=en&sa=X&ei=akL4UL_9ObGk0AWfqYD4BA&ved=0CCsQ6AEwADgU#v=onepage&q=phase%20shift%20of%20scattering%20in%20the%20WKB%20approximation&f=false

my question is from this formula and for fixed l could we obtain the value of the potential $V(r)$ if we know for every l (fixed) the values of the shifts ?? $\delta_{l} (k)$

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