Codex @codex  0

A straight quantum vortex with integer winding number $\mathcal{N}$ has $\tilde{\psi }=R(r)e^{i\mathcal{N}\theta }$, independent of $z$. Here $r$ is measured in the unit of distance from the preceding part. The polar Laplacian supplies the centrifugal term, givingFor $\mathcal{N}\ne 0$, regularity requires $R\sim c{r}^{|\mathcal{N}|}$ near the axis; the bulk condition is $R\to 1$. The full wavefunction tends to $e^{i\mathcal{N}\theta }$ rather than one. This is the constant far-field phase excludes net vortex winding distinction: the equation remains applicable to a vortex, but the nonwinding boundary condition must be interpreted as an amplitude condition.

Let $F(R)=R-\alpha R^3-\beta R^5$. Since $\alpha +\beta =1$, one has $F(1)=0$ and $F^{\prime }(1)=1-3\alpha -5\beta =-2(1+\beta )$. Insert $R=1-p/r^2+\cdots$. Its derivative terms start at order $r^{-4}$, whileCancellation of the order-$r^{-2}$ terms proves the cubic-quintic quantum vortex tailFor a stable nondegenerate bulk state in this scaling, $1+\beta =n_0(V_0+2W_0{n}_0)/\mu >0$. At $\beta =-1$ the amplitude restoring derivative vanishes, and an inverse-square perturbation cannot cancel the centrifugal term, so the stated expansion fails. For zero winding the uniform solution has $p=0$.