Remember that in the tortoise coordinate, r ∗ ∗ → → − − ∞ ∞ denotes event horizon and r ∗ ∗ → → ∞ ∞ is equivalent to r → → ∞ ∞ i.e., to distances far away from the back hole. The potentials are short ranged as they decay faster than 1 / r ∗ ∗ ; as r ∗ ∗ → → ∞ ∞ we have V ± ± → → 2 ( n + 1 ) / r 2 and as r ∗ ∗ → → − − ∞ ∞ , we have V ± ± ∼ ∼ e r ∗ ∗ / 2 M . Consequently, the asymptotic behaviour of the solutions for r ∗ ∗ → → ± ± ∞ ∞ is e ± ± i σ σ r ∗ ∗ .