Return to Revisions

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.