Zero-energy solutions and vortices in Schrödinger equations
Zero-energy solutions and vortices in Schrödinger equations
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 …