What are the boundary conditions at $r=0$ for Schr$\ddot{\textrm{o}}$dinger's equation for a quantum particle in 2D polars $(r,\theta)$, with potential $U=0$ for $r and $U=\infty$ for $r>a$?
I've shown that the angular part of the wave-function is $e^{im\theta}$ where $m\in\mathbb{Z}$, and I've got an ODE for the radial part $R(r)$, with boundary condition $R(a)=0$.
Since $e^{im\theta}$ is not well-defined as $(r,\theta)$ approaches the origin, it looks as though we should have $R(0)=0\;$ to get a well-defined wave-function. But according to the question, the series expansion for $R(r)$ has arbitrary constant term and all odd-order terms zero. What sort of boundary conditions give this??
Many thanks for any help with this!