_{nk}=\epsilon _{n}\Psi _{nk},\qquad n=0,1,2,...\text{ }and\text{ }%
k\in {\cal Z}  \eqnum{2.10}$$

has eigenvalues and eigenfunctions given respectively by

$$\epsilon _{n}=\left( n+\frac{1}{2}\right) \omega _{c}\qquad  \eqnum{2.11}$$

$$\Psi _{nk}=\frac{\overline{b}^{1/4}}{\sqrt{L}}\exp \left( -2\pi
ikx_{1}/L\right) \varphi _{n}\left( x_{2}\sqrt{\overline{b}}-\frac{2\pi k}{L%
\sqrt{\overline{b}}}\right)  \eqnum{2.12}$$

where $\omega _{c}=\overline{b}/m$ is the cyclotron frequency and $\varphi
_{n}\left( \xi \right) $ are the orthonormalized harmonic oscillator wave functions.

Note that the energy levels $\epsilon _{n}$ are degenerates (they do not depend on $k$). Then, for each Landau level $n$ there exists a band of degenerate states. The cyclotron frequency $\omega _{c}$ plays here the role of the energy gap between occupied Landau levels. It is easy to prove that the filling factor, defined as the ratio between the density of particles $%
n_{e}$ and the number of states per unit area of a full Landau level, is equal to the Chern-$%
%TCIMACRO{\func{Si}}
%BeginExpansion
\mathop{\rm Si}%
%EndExpansion
$mons coupling constant $N$. Thus, because we are considering that $N$ is a positive integer, we have in this MCS theory $N$ completely filled Landau levels. Once this ground state is established, it can be argued immediately [@6], [@6b], [@10a], [@15], that at $T=0$ the system will be confined to a filled band, which is separated by an 