Consider the following ODE eigenproblem of $y(x)$
\begin{equation}
 y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0
\end{equation}
with eigenvalue $\varepsilon$, real constants $a,b$. The boundary condition is $y(\pm\infty)=0$. Numerically, this turns out to have well-behaved eigensolutions.


My question is how to see the typical length scale of the eigensolution $y(x)$, i.e., how it asymptotically decays. For instance, if $y(x)\sim e^{-x^2/c^2}$, $c$ is the length scale I mean.




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This ODE can also be shown to have the following general solution
\begin{equation}
 y(x)= \sum\_{s=\pm} C\_s\, e^{-arx\_s - \frac{x\_s^3}{2}} \mathscr{H}\_\mathrm{T}(\alpha,\beta\_s,\gamma,x\_s)
\end{equation}
with integration constants $C\_\pm$, $r=(\frac{3}{b^2})^{\frac{1}{3}}$, $\alpha=r^2\varepsilon,\beta\_\pm=\pm3,\gamma=2ra,x\_\pm=\pm x/r$ and $\mathscr{H}\_\mathrm{T}$ the [triconfluent Heun's function](https://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunT). However, its asymptotics is not solely determined by the exponential factor, because $\mathscr{H}\_\mathrm{T}$ is not truncated to be a finite polynomial for these $\beta$'s, although overall $y(x)$ decays well. So it's not clear to me whether this general solution helps the above question.



Asymptotic behavior of an ODE
The quick-and-dirty way to guess the asymptotic behavior is to substitute in a WKB ansatz $y = e^{S(x)}$ and keep only the leading order terms, which here would be the highest power of $x$ and the highest power of $S'(x)$, namely $(S')^2 - (x^2 b^2/2)^2 = \text{l.o.t}$. The solution is $S(x) \sim \pm x^3 b^2/6$ as $|x| \to \infty$. This is exactly the leading asymptotic for $S(x)$ captured by your general solution.


One can go through a longer algorithmic procedure (for any singular ODE with meromorphic coefficients, of which yours is a special case) to get very precise information about the full asymptotic expansion. I give a reference and go through an example calculation for a different equation in [this answer](https://mathoverflow.net/a/282446).


