Interacting gauge fields and the zero-energy eigenstates in two   dimensions
  Gauge fields are formulated in terms of the zero-energy eigenstates of 2-dimensional Schr$\ddot {\rm o}$dinger equations with central potentials $V_a(\rho)=-a^2g_a\rho^{2(a-1)}$ ($a\not=0$, $g_a>0$ and $\rho=\sqrt{x^2+y^2}$). It is shown that the zero-energy states can naturally be interpreted as a kind of interacting gauge fields of which effects are solved as the factors $e^{ig_c\chi_A}$, where $\chi_A$ are complex gauge functions written by the zero-energy eigenfunctions. We see that the gauge fields for $a=1$ are nothing but tachyons that have negative squared-mass $m^2=-g_1$. We also find out U(1)-type gauge fields for $a=1/2$ and SU(3)-type gauge fields for $a=3/2$. Massive particles with internal structures described by the zero-energy states are also studied. 
Tsunehiro Kobayashi