First of all, it is sufficient to consider only one parameter: making a change of the independent variable $z\mapsto kz$, with appropriate $k$ one can eliminate either $a$ or $b$. Let us eliminate $b$ and consider
$$-y''+(x^4+a^2x^2)y=\lambda y.$$
Then eigenvalues (in $L^2(R)$) become functions of $a$, and the asymptotics is
$$\lambda_n\sim cn^{4/3},$$
where $c$ is an absolute constant (it does not depend on $a$). One can write several terms of asymptotic expansion in decreasing powers of $n$, coefficients of these
further terms will depend on $a$. In fact one can write and prove an infinite
asymptotic expansion. These functions were very much studied, I mention one paper:

C. Bender and T. Wu, Anharmonic oscillator. Phys. Rev. (2) 184 1969
1231–1260.

This paper has more than 1000 references on Google scholar. By looking in these references you can obtain a more or less complete picture of what is known.
(The situation is roughly like this: all statements in this paper are correct,
and it contains a very complete discussion of these eigenvalues. 
However most of the things are proved of "physical level of rigor", or just illustrated by computation. In the subsequent papers most
of these statements were rigorously justified.)