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 $a$. These functions were very much studied, I mention one paper:
C. Bender and T. Wu, Anharmonic oscillator. Phys. Rev. (2) 184 1969 1231–1260.
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