I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter the equation as:
$$ \frac{d}{dz}\left( (1-z^2)\frac{dw}{dz}\right)+\left( \lambda + \gamma^2(1-z^2)-\frac{\mu^2}{1-z^2} \right)w=0, $$
are determined by requiring regularity at the boundary points $z=\pm1$, which are singular (reference). This constrains them to take discrete values, e.g. for $\gamma=0$ they would just be $\lambda_l = l(l+1)$, since the spheroidal equation reduces to the associated Legendre equation. My question is the following: if, for some reason, we decided to shift the boundaries of the problem so that we are only including one of the singular points in our domain (say $z\in [-0.5,1.5]$), how would that affect the allowed eigenvalues for the solution? For the case $\gamma=0$, would you still be able to claim that $\lambda_l = l(l+1)$, or would you have insufficient information? Thanks in advance.