Eigenstates of Conical Potential in 3-dimensions?

Question

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.

Answer 1

See this answer of mine. In $d$ spatial dimensions, your equation reads $$ u''(r)+2m[E-V_\ell(r)]u(r)=0 $$ where the effective potential is $$ V_\ell=V(r)+\frac{1}{2m}\frac{\ell_d(\ell_d+1)}{r^2} $$ with $\ell_d=\ell+(d-3)/2$. The zero-angular momentum state has $\ell=0$, and therefore in $d=3$ dimensions the equation for $u(r)$ is identical to the 1D Airy equation, whose solution you already know. For $\ell_d\neq 0$ there doesn't seem to be analytical solutions. The asymptotic behaviour at $r\to\infty$ should be easy to calculate, inasmuch the centrifugal term is negligible as compared to the linear term $Ar$. Other properties of the system are not as easily estimated using analytical methods, but one can always resort to numerical methods.