# Spheroidal eigenvalues with shifted boundary conditions

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.

• Yes, if the $z\equiv\cos\theta=\pm1$ axes are not included in the region in which you are trying to find the solution, you do indeed need to include the possibility of solutions that may be irregular on the axis. There is an an example of this in Jackson, section 3.4, calculating the fields inside a cone. The conical boundary imposes a quantization condition on $\lambda$, but it is not the usual $\lambda=\ell(\ell+1)$.
– Buzz
Oct 5, 2023 at 22:29