Self-adjoint extension for Maxwell-Chern-Simons model in long wavelength limit

In the long wavelength limit, Maxwell-Chern-Simmon model and the dynamics of a particle in a plane under an external magnetic field perpendicular to that plane are identical. The self adjoint extension of such a problem depends on the value of angular momentum quantum number $l$. In this article, we have shown that for $l\neq 0$, the operator describing the Landau level wave-function is self adjoint; whereas, for $l=0$, infinite number of self-adjoint extension by an one parameter unitary mapping is possible.