I am trying to find eigen-values, $E$, for the following differential operator: $$\left[ -\frac{1}{2}\frac{d^2}{dx^2} +L\left(x^2-a^2\right)^2\right]y(x) = E\,y(x) $$
where $L,a$ are two positive real parameters, given the boundary conditions that $\lim_{|x|\to\infty}y(x)=0$. I am only looking for eigenvalues which are smaller than $La^4$.
What I have tried:
Plug in a power series $y(x)=\sum_{n=0}^{\infty}\alpha_n(x-a)^n$ and get a recurrence relation on the coefficients: \begin{align} \alpha_2 &= -2E\alpha_0 \\ \alpha_3 &= -\frac{1}{3}E\alpha_1 \\ \alpha_4 &= -\frac{1}{3}(E^2-2La^4)\alpha_0 \\ \alpha_5 &= (-\frac{1}{30}E^2+\frac{4}{10}a^2)\alpha_1+\frac{4}{10}L\,a\,\alpha_0 \\ \alpha_{n+2} &= \frac{-2E\alpha_n+8La^2\alpha_{n-2}+8La\alpha_{n-3}+2L\alpha_{n-4}}{(n+2)(n+1)} \forall\,n\in\left\{4,\,5,\,6,\,\dots\right\} \end{align} However, I wasn't really sure how to proceed from there. I am also unsure how the boundary conditions apply to finding $\alpha_0$ and $\alpha_1$.