# Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form

$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$

for certain constants $c_{1}\in\mathbb{C}$ with $\textrm{Re}(c_{1})=-2$ and $c_{2}\geq 0$. Does anyone know whether such an operator has been studied before, and if so, where we could find information about it? Specifically, we would like to know what the eigenvalues of $L$ as an operator on $L^{p}(\mathbb{R})$ for $p\in(1,\infty)$ are. In other words, we are interested in solutions $f$ to the equation

$-(1+x^{2})\frac{d^{2}f}{dx^{2}}+c_{1}x\frac{df}{dx}+c_{2}x^{2}f-c_{3}f=0$

for which $f\in L^{p}(\mathbb{R})$ and $c_{3}\in\mathbb{C}$.

The constants $c_{1}$ and $c_{2}$ will vary within the range prescribed, but information for specific values of the constants would also be welcome. Any information would be much appreciated!