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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!

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    $\begingroup$ This equation has regular singular points at $\pm i$ and an irregular singular point of rank 1 at infinity. This makes it a confluent Heun equation. This should help get you started on a literature search. You could also exploit symmetry and substitute $u=x^2$. $\endgroup$
    – Michael Renardy
    Commented Jan 20, 2014 at 18:20

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Using Maple, the general solution of the ODE is: $$ f(x) = k_1\cdot(x^2+1)^{1+(1/2) c1}\cdot \mathrm{HeunC}(0, -1/2, 1+(1/2)c1,(1/4)c2, (1/4)c3+(1/8)c1+1/2,-x^2)+k_2\cdot (x^2+1)^{1+(1/2)c1}\cdot x\cdot\mathrm{ HeunC}(0, 1/2, 1+(1/2)c1, (1/4)c2, (1/4)c3+(1/8)c1+1/2, -x^2)$$

Now one has to show that there are $k_1$ and $k_2$ such that $f$ is in $L^p$.

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