I would like to know if the following differential operator $L$ on $(0,\infty)$ is well-known or derived from such one: \begin{align} L := \frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx} \quad (a,b > 0). \end{align} I also would like to know if the eigenfunction (i.e., a function $f$ on $(0,\infty)$ s.t. $Lf(x) = \lambda f(x)$ for each $\lambda \in \mathbb{R}$) is represented by some special functions.
When $b = 1$, by changing variables as $y = \sqrt{2x}$, the operator $L$ is transformed to \begin{align} y \frac{d^{2}}{dy^{2}} + \left( \frac{1}{2} - ay \right)\frac{d}{dy}, \end{align} and thus the eigenfunctions are given by the confluent hypergeometric functions. But for $b \neq 1$, I can't reduce the operator to a well-known one.
Thank you for any comments.