I was trying to solve this differential equation: $$ \frac{\partial^2 \phi}{\partial t^2} - c^2\left( 1- \frac{a}{r}\right )\nabla^2 \phi = 0 $$ $$ a \in \mathbb R, \; \; \; t > 0, \; \; \; r > 0 $$ Assuming no angular dependence: $$ \phi(r, t) = R(r)T(t) $$ $$ T(t)=e^{\pm i \omega t} $$
$$ \left ( 1- \frac{a}{r}\right )\left ( \frac{d^2R}{dr^2} + \frac{2}{r} \frac{dR}{dr} \right) +k^2 R=0 $$ where $\omega/k = c$. Assuming a solution of the form: $$ R(r) = \frac{\varphi(r)}{r} $$ reduces the equation to: $$ (r-a)\frac{d ^2 \varphi}{dr^2} + k^2 r \varphi =0 $$ Working out a solution (asking Wolfram) I get that: $$ \varphi(r) = L^{-1}_{iak/2} \left ( 2ik(a- r) \right )e^{-ik(a-r)} $$ Where: $$ L^{(m)}_{n}(x) $$ are the asociated Laguerre polynomials. The thing is, what is the meaning of: $$ n \in \mathbb C $$ in this case: $$ n = \frac{iak}{2} $$ Why the imaginary unity appears there? Is there any way to rewrite the solution in terms of real valued functions? Is there any "more elegant" or clean way of writting the solution to the differential equation?
