I would like to solve the functional differential equation $$ y''(x) - A y'(x) - B y(x) + \alpha B y(\alpha x)= 0$$ for $-\infty < x < \infty$ and boundary conditions $y(x \rightarrow \pm \infty) = 0$. Here the parameter $\alpha > 1$. I am unfamiliar with standard approaches to such functional differential equations and guidance of any kind is most welcome!
Progress: So far, I found the Dirichlet series solution (which is suggested by Laplace transforms) $$y(x) = \sum_{k=0}^\infty C_k e^{-\alpha^k r x }$$ solves the equation but not the boundary conditions. Using this ansatz provides the recursion relation $$ C_k(r^2\alpha^{2k} + A r \alpha^k - B) = -\alpha B C_{k-1}$$ for the coefficients in the series, and taking $k=0$ gives a characteristic equation for $r$ $$ r_\pm = \frac{A}{2}\Big(-1 \pm \sqrt{1 + 4B} \Big).$$ Thus the general solution has the form $$y(x) = K_+ \sum_{k=0}^\infty \frac{(-\alpha B)^k e^{-\alpha^k r_+ x}}{\prod_{l=1}^k(r_+^2\alpha^{2l} + A r_+ \alpha^l - B)} + K_- \sum_{k=0}^\infty \frac{(-\alpha B)^k e^{-\alpha^k r_- x}}{\prod_{l=1}^k(r_-^2\alpha^{2l} + A r_- \alpha^l - B)}$$ which is a super interesting and nontrivial structure. The $K_\pm$ are constants to be set by the boundary conditions.
Here's my confusion: The series involving $r_+$ only converges when $x>0$, while the series involving $r_-$ only converges when $x<0$. How can I incorporate these facts into my solution? Can I drop the divergent solutions to get a two-sided solution
$$y(x) = K \theta(x) \sum_{k=0}^\infty \frac{(-\alpha B)^k e^{-\alpha^k r_+ x}}{\prod_{l=1}^k(r_+^2\alpha^{2l} + A r_+ \alpha^l - B)} + K\theta(-x) \sum_{k=0}^\infty \frac{(-\alpha B)^k e^{-\alpha^k r_- x}}{\prod_{l=1}^k(r_-^2\alpha^{2l} + A r_- \alpha^l - B)}?$$ The parameter $K$ can be set by a probabilistic normalization condition in context of my problem.
Is this application of boundary conditions valid? In a way it seems contrived since I'm allowing the $K_\pm$ constants to depend on $x$.