Boundary conditions for a step potential

[Dario SLC]

Homework Statement

A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height $V_0$.
What is the boundary conditions for this eigenfunctions?
Find the degeneracy level for the energy, when it is $E<V_0$

Homework Equations

Radial equation
\begin{equation}
-\frac{\hbar^2}{2m}\left(\frac{\partial^2u}{\partial r^2}-\frac{l(l+1)}{r^2}u\right)+V(r)u(r)=Eu(r)
\end{equation}
with $u(r)=rR(r)$

The Attempt at a Solution

Well, first the constant of motion it $\hat{L_z}$, $\hat{L^2}$ and $\hat{S_z}$ because conmute with $\hat{H}$, also $\hat{H}$, and it former a complete set of observables that conmute, therefore the form of eigenfunctions are:
\begin{equation}
\psi(r,\theta,\varphi,\sigma)=R(r)Y(\theta,\varphi)\chi(\sigma)
\end{equation}
when $\chi(\sigma)$ is a spin function, and only take the values $m_s={+1/2,-1/2}$ because eigenvalues of operator $\hat{S^2}$ is $3/4\hbar^2$ with $s=1/2$.

Here my doubt:
Boundary conditions to $Y(\theta,\varphi)$ this is referred to:
$l=0, 1, \dots, n-1$ and $-l\leq m\leq +l$?

For radial function, it must be:
because $\hat{H}$ is hermitian, then for $r=0\Longrightarrow u(0)\longrightarrow0$
and for $r\longrightarrow\infty \Longrightarrow u\longrightarrow0$

For $\chi$ the boundary conditions it is $-s\leq m_s\leq+s$?

In the case for the degeneracy, when $E<0$ corresponds to $l=0$ because it is a bound state, then without spin the level of degeneracy is $n^2$ but with spin is $2n^2$.

Up to here my poor attemp to solution, i am really puzzled for explain the boundary condition to Y and $\chi$, and i am not sure that the
$r\longrightarrow\infty \Longrightarrow u\longrightarrow0$ for this potential.

 

[hilbert2]

The boundary conditions are that the wavefunction $u(r)$ and its radial derivative $\frac{\partial u}{\partial r}$ must be continuous at the potential step, and for bound states $u$ and $\frac{\partial u}{\partial r}$ must approach zero when going to $r\rightarrow\infty$ (for unbound states they must only remain finite). What do you know about the degeneracy of hydrogen atom energy eigenstates? The idea is the same for all spherically symmetric potentials, not only $V\propto r^{-1}$