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?



Eigenvalue problem of Schrodinger equation with polynomial growth potential on the real line
If $L=-\Delta +|x|^s$, $s>0$, in $R^n$, then $N(\lambda)$, the number of eigenvalues less than $\lambda$, behaves like $\lambda^{N(1/2+1/s)}$ as $\lambda \to \infty$. This is in Titchmarsh "Eigenfunction expansions...Part II, Sect. 17.8 or in Reed Simon Section XIII.


