Zero-energy solutions and vortices in Schrödinger equations

Abstract

Two-dimensional Schr\"odinger equations with rotationally symmetric potentials $[{V}_{a}(\ensuremath{\rho})=\ensuremath{-}{a}^{2}{g}_{a}{\ensuremath{\rho}}^{2(a\ensuremath{-}1)}$ with $\ensuremath{\rho}=\sqrt{{x}^{2}{+y}^{2}}$ and $a\ensuremath{\ne}0]$ are shown to have zero-energy states. For the zero energy eigenvalue the equations for all a are reduced to the same equation representing two-dimensional free motions in the constant potential ${V}_{a}=\ensuremath{-}{g}_{a}$ in terms of the conformal mappings of ${\ensuremath{\zeta}}_{a}{=z}^{a}$ with $z=x+iy.$ Namely, the zero-energy eigenstates are described by plane waves with the fixed wave numbers ${k}_{a}=\sqrt{{2mg}_{a}}/\ensuremath{\Elzxh}$ in the mapped spaces. All the zero-energy states are infinitely degenerate similar to the case of the parabolic potential barrier (PPB) shown by Shimbori and Kobayashi [J. Phys. A 33, 7637 (2000)]. Following hydrodynamical arguments, we see that such states describe stationary flows around the origin, which are represented by the complex velocity potentials ${W}_{a}=\sqrt{{2g}_{a}/m}{z}^{a},$ and their linear combinations create almost arbitrary vortex patterns. Examples of the vortex patterns in constant potentials and PPB are presented. In the extension to three-dimensional problems with potentials being separable into $2+1$ dimensions we show that the states in three dimensions have the same structure as the two-dimensional states with the zero energy but they can generally have nonzero total energies.