Boundary conditions of the radial Schrodinger equation

[spaghetti3451]

Consider the radial differential equation

$\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)$,

which I've obtained by solving the Schrödinger equation in $d$ dimensions using the method of separation of variables.

Now, the boundary condition that I have been given is $\phi_l (r) \sim r^{l+\frac{d-1}{2}}, r \rightarrow 0$.

However, I was expecting the boundary conditions $\phi_l (0) = 0, \phi^{'}_{l} (0) = 1$.

Does anybody have an idea if there is a relation of some sort between the two sets of boundary conditions in the context of the given differential operator.

 

[ShayanJ]

The first relation isn't a boundary condition! What it means, is that we let r approach zero and see what will be the solution to the limiting equation. Then we demand that the solution to the full equation approaches the solution to the limiting equation. You can do a similar thing for $r\rightarrow \infty$. This is called checking the asymptotic behaviour of the equation and is a means of gaining more information about the solution of the equation so that we can solve it easier. Boundary conditions only come into play after we have found the solution of the equation.