I'm working on obtaining the spectrum of the Klein-Gordon operator in Hyperbolic space $H_2$ in a given coordinate system, but I'm having trouble solving for the continuous part of the spectrum.
The eigenvalue differential equation reads \begin{equation} -\sinh^2(x) \left[\frac{d^2}{dx^2} - \left(k - \frac{B}{\tanh(x)} \right)^2 \right]\phi(x) = \lambda \phi(x) \end{equation}
I managed to obtain the discrete spectrum through a variable change \begin{equation} y = \coth(x) \end{equation} Followed by substituting the solution \begin{equation} \phi(y) = (1-y)^\alpha(1+y)^\beta g(y) \end{equation} Which resulted in the Hypergeometric differential equation, and eigenvalues $\lambda$ with a dependency on $a \in \mathbb{N}$, from $\textbf{F}(a,b;c;x)$.
However, I'm having trouble finding the path to obtain a spectrum that involves a continuous variable, namely, I tried to reach the hypergeometric equation in a way that would not involve the coefficient $a$ being an integer, or tried transforming it into the Whittaker equation. It may help to know that in the small limit $x \to 0$, the eigenfunctions(for the continuous part) should reduce to Whittaker functions, with eigenvalues: \begin{equation} \lambda = \frac{1}{4} + B^2 + \nu^2, \quad 0 \leq \nu < \infty \end{equation}
All help is appreciated.
Thank you for your attention.