I am trying to find eigenfunctions (if there are any to be found!) of the operator:
$$T\left\{ f\right\} \left(x\right)=\frac{2}{9}f\left(-a-\frac{9}{2}x\right)+\frac{2}{9}f\left(b-\frac{9}{2}x\right)$$
where $a$ and $b$ are positive rational numbers.
Setting up the eigenfunction equation and performing some simple changes of variables produces:
$$\frac{9\lambda}{2}f\left(-\frac{2}{9}\left(x+\alpha\right)\right)=f\left(x+\beta\right)+f\left(x-\beta\right)$$
where $$\alpha=\frac{a-b}{2}, \beta=\frac{a+b}{2},$$ and where $\lambda$ is the eigenvalue.
(Note: there is a simple analytic solution when $\lambda=0$.)
Any ideas, people?