Eigenfunctions in Spherically Symmetric Well
I am looking at a problem that has a potential
$$ V(r) = \begin{cases} 0 & a<r<b\ \infty & \text{elsewhere} \end{cases} $$
This is a modification of the infinite spherical well problem, since the well does not extend to $r = 0$. The eigenfunctions are of the form $\Psi(r,\theta,\phi) = R(r)Y_{l,m}(\theta,\phi)$, where the $Y_{l,m}$'s are the usual spherical harmonics. From this, it can be shown that the radial equation becomes
$$ \frac{d^2R}{d\rho^2} + \frac{2}{\rho}\frac{dR}{d\rho} + \bigg[1-\frac{l(l+1)}{\rho^2}\bigg]R = 0, $$ where $\rho\equiv kr$ is a dimensionless parameter, and $k^2 \equiv 2mE/\hbar^2$. This equation is the spherical <PERSON> equation, and has known solutions that are the spherical <PERSON> (regular) and spherical <PERSON> (irregular) functions. The <PERSON> functions are regular in that they go to zero as $r$ goes to zero, whereas the <PERSON> functions do not.
This is where I am stuck; can we in this case assume that the solution are just the regular Bessel functions? In the infinite spherical well, where have $V(r) = 0$ for $0<r<a$, so we must have regularity at the origin. In that case the radial functions would be $$ j_l(\rho) = (-\rho)^l \bigg(\frac{1}{\rho}\frac{d}{d\rho}\bigg)^l\frac{\sin\rho}{\rho} $$ So, for example, the ground state function, i.e. 474
$j_0(\rho) = \sin\rho/\rho$. Applying the boundary condition at $r = a$, i.e. $j_0(\rho = ka) = 0$ gives $$ \frac{\sin ka}{ka} = 0 \Rightarrow ka = n\pi $$ Substituting the expression for $k$, we ultimately get $$ E_{n,l=0} = \frac{\pi^2\hbar^2}{2ma^2}n^2 $$
This is all very well known. However, in the problem at hand, we also require that $R(\rho = kb) = 0$ in addition to $R(\rho = ka) = 0$. Thus, I am not sure that we can just use the regular Bessel functions are eigenfunctions. I suppose you can just make the substitution $a \rightarrow b-a$ so that the ground state energies are $$ E_{n,l=0} = \frac{\pi^2\hbar^2}{2m(b-a)^2}n^2, $$ but this is not rigorous at all and seems fishy to me. My question is then: How would you write the radial eigenfunctions in this case, and subsequently how to get the energy levels using both boundary conditions? 474