Asymptotic study of eigenvalues

Question

I have the differential equation $$(a + x - b(1+x^2))\frac{dg}{dx}(x)+(1+x^2)\frac{d^2g}{dx^2}(x) + \lambda g(x)=0 \, $$ where $a, b >0$ and $x$ is a real variable. I want to find out the bounded states i.e. I want to find $\lambda_n$ such $$\lim_{x\rightarrow \infty} e^{-bx} g(x) \rightarrow\ 0 \, . $$ I think there is no closed form for $\lambda_n$, therefore I am trying to find out some expansion in $b$ for $b\rightarrow \infty$, i.e. I am looking for $$\lambda_n = \sum_{k=0}^{\infty}\lambda_{nk}(a)b^{-k} \, .$$ Does someone have any idea how I could go about this problem and find out at least the first order. Any literature I could look into or any articles related. Thank you!