Zero Energy Solutions and Vortices in Schroedinger Equations

Abstract

All two-dimensional Schrödinger equations with symmetric potentials \break $(V_a(\rho )=-a^2{g}_a{\rho }^{2(a-1)/2}with\rho =\sqrt{x^2+y^2}anda\ne 0)$ is shown to have zero energy states contained in conjugate spaces of Gel'fand triplets. 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 $V_a=-g_a$ in terms of the conformal mappings of ${\zeta }_a=z^a$ with $z=x+iy$. Namely, the zero energy eigenstates are described by the plane waves with the fixed wave numbers $k_a=\sqrt{m{g}_a}/\hslash$ in the mapped spaces. All the zero energy states are infinitely degenerate as same as the case of the parabolic potential barrier (PPB) shown in ref. \cite{sk4}. Following hydrodynamical arguments, we see that such states describe stationary flows round the origin, which are represented by the complex velocity potentials $W=p_a{z}^a$, ($p_a$ being a complex number) and their linear combinations create almost arbitrary vortex patterns. Examples of the vortex patterns in constant potntials and PPB are presented.

12 pages, 4 figures

References in corpus (1)

• Vortex Lattices in Quantum Mechanics

Cited by in corpus (6)

• Vortex Lattices in Quantum Mechanics
• L2 series solutions of the Dirac equation for power-law potentials at rest mass energy
• Symmetry breaking and singularity structure in Bose-Einstein condensates
• New Classes of Potentials for which the Radial Schrodinger Equation can be solved at Zero Energy
• Quantum Mechanics of Particles Constrained to Spiral Curves with Application to Polyene Chains
• Zero-Energy Flows and Vortex Patterns in Quantum Mechanics