A particle moves in a central potential U(r) = −k/re^−r/ξ (screened potential, with k 0 and ξ 0). For what values of the angular momentum l_0 a bounded motion of the particle (i.e. when the particle does not escape to the infinity) is possible?

Question

A particle moves in a central potential U(r) = −k/re^−r/ξ (screened potential, with k > 0 and ξ > 0). For what values of the angular momentum l_0 a bounded motion of the particle (i.e. when the particle does not escape to the infinity) is possible?
Hint: A bounded motion is possible if the effective potential has a minimum – you can admit that if the effective potential has an extremum, then this is a minimum. You might find it useful to introduce the dimensionless variable
u = r/ξ, and to study the function f(u) = u(1 + u)e^−u.