Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem $$L\phi=\lambda\phi$$ where $\phi$ is in $L^2(\mathbb{R})$ and vanishes at infinity.

We want to study the distribution of the eigenvalues (approximately)?

Classical example: when $s=2$, Hermite orthogonal polynomial comes out.

Is there any analytical result about $s=4$ case?

I am currently looking at books by Titchmarsh , and ODE book by Coddington and Levinson.

Any other classical/modern books papers for reference?