Bound states in a 2D short range potential induced by spin-orbit interaction

TL;DR

This paper reveals that a 2D short-range potential with spin-orbit interaction hosts infinitely many bound states, contrasting with the single bound state in the absence of SO coupling, especially for shallow wells.

Contribution

It demonstrates that spin-orbit interaction induces an infinite number of bound states in a 2D short-range potential, a novel finding in quantum bound state theory.

Findings

Infinite bound states appear due to SO interaction in 2D potentials.

For shallow wells, only one degenerate bound state exists per half-integer angular momentum.

Binding energy decreases rapidly with increasing angular momentum quantum number.

Abstract

We have discovered an unexpected and surprising fact: a 2D axially symmetric short-range potential contains {\it infinite} number of the levels of negative energy {\it if one takes into account the spin-orbit (SO) interaction.} For a shallow well ($m_eU_0R^2/\hbar^2 \ll 1$, where $m_e$ is the effective mass, $U_0$ and $R$ are the depth and the radius of the well, correspondingly) and weak SO coupling ($|\alpha|m_eR/\hbar \ll 1$, $\alpha$ is the SO coupling constant) exactly one two-fold degenerate bound state exists for each value of the half-integer moment $j=m+1/2$, and the corresponding binding energy $E_m$ extremely rapidly decreases with increasing $m$.