If we take an ordinary time-invariant Schödinger equation: $$H|\psi\rangle = E|\psi\rangle,$$ and use a conical potential $V(r) = A r$ we get a differential equation: $$\left[-\left(\frac{\hbar^2}{2m}\right)\nabla^2 + A r\right]\psi\left(\vec{r}\right) = E\psi\left(\vec{r}\right).$$

In one dimensions this becomes the Airy differential equation, with the Airy function, $\psi_n(x) = N \operatorname{sgn}(x-x_0)\operatorname{Ai}(k|x-x_0|)$, giving normalizable solutions for values of $x_0$ fixed by the energy and inverse scale $k=\sqrt[3]{\frac{2Am}{\hbar^2}}$.

Are the eigenvalues and eigenstates known for the 2 and, especially, 3 dimensional cases? Even for the zero angular momentum states? I've asked about whether the resulting differential equation can be related to a standard one with known solutions over at math.stackexchange and don't have a response there, but I thought that someone here at physics.stackexchange might have a greater familiarity with this particular problem.

If we take an ordinary time-invariant Schödinger equation: $$H|\psi\rangle = E|\psi\rangle,$$ and use a conical potential $V(r) = A r$ we get a differential equation: $$\left[-\left(\frac{\hbar^2}{2m}\right)\nabla^2 + A r\right]\psi\left(\vec{r}\right) = E\psi\left(\vec{r}\right).$$

In one dimensions this becomes the Airy differential equation, with the Airy function, $\psi_n(x) = N [\operatorname{sgn}(x-x_n)]^n \operatorname{Ai}(k|x-x_n|)$, giving normalizable solutions for values of $x_n$ fixed by the energy and inverse scale $k=\sqrt[3]{\frac{2Am}{\hbar^2}}$.

Are the eigenvalues and eigenstates known for the 2 and, especially, 3 dimensional cases? Even for the zero angular momentum states? I've asked about whether the resulting differential equation can be related to a standard one with known solutions over at math.stackexchange and don't have a response there, but I thought that someone here at physics.stackexchange might have a greater familiarity with this particular problem.