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

## TL;DR

This paper investigates the self-adjoint extensions of the Maxwell-Chern-Simons model in the long wavelength limit, revealing how angular momentum quantum numbers influence the mathematical properties of the associated operators.

## Contribution

It demonstrates the conditions under which the Landau level operator is self-adjoint, especially highlighting the case of zero angular momentum quantum number.

## Key findings

- For l ≠ 0, the operator is self-adjoint.
- For l = 0, multiple self-adjoint extensions exist, parametrized by a unitary mapping.
- The analysis connects the mathematical structure to physical quantum states.

## Abstract

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.