Stationary Flows of the Parabolic Potential Barrier in Two Dimensions

Abstract

In the two-dimensional isotropic parabolic potential barrier $V(x, y)=V_0 -m\gamma^2 (x^2+y^2)/2$, though it is a model of an unstable system in quantum mechanics, we can obtain the stationary states corresponding to the real energy eigenvalue $V_0$. Further, they are infinitely degenerate. For the first few eigenstates, we will find the stationary flows round a right angle that are expressed by the complex velocity potentials $W=\pm\gamma z^2/2$.