Physics – Condensed Matter – Mesoscale and Nanoscale Physics
Scientific paper
2001-03-09
Physics
Condensed Matter
Mesoscale and Nanoscale Physics
12 pages, 4 figures
Scientific paper
10.1103/PhysRevA.65.042108
All two-dimensional Schr\"{o}dinger equations with symmetric potentials \break $(V_a(\rho)=-a^2g_a \rho ^{2(a-1)/2} {with} \rho=\sqrt{x^2+y^2} {and} a\not=0)$ 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 $k_a=\sqrt{mg_a}/\hbar$ 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.
Kobayashi Tsunehiro
Shimbori Toshiki
No associations
LandOfFree
Zero Energy Solutions and Vortices in Schroedinger Equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Zero Energy Solutions and Vortices in Schroedinger Equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Zero Energy Solutions and Vortices in Schroedinger Equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-308102