spatial trace, $$Z(\beta) = \int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_2\;
    K({\bf r},{\bf r};-i\hbar\beta)\; .$$ Substituting (\[PropagadorFinalBUniforme\]) into last expression, we get $$Z(\beta) = \int_{-\infty}^\infty dx_1\int_{-\infty}^\infty dx_2\;
    {m\omega\over 2\pi\hbar\,\mbox{senh}(\hbar\beta\omega)}\; ,$$ where we used the fact that $\sin(-i\theta)=-i\,\sinh\,\theta$. Observe that the above result is divergent, since the area of the ${\cal OX}_1{\cal X}_2$ plane is infinite. This is a consequence of the fact that each Landau level is infinitely degenerated, though the degeneracy per unit area is finite. In order to proceed, let us assume an area as big as we want, but finite. Adopting this kind or regularization, we write $$\begin{aligned}
    \int_{-L/2}^{L/2} \!\! dx_1\!\!\int_{-L/2}^{L/2}\!\! dx_2\;
    \!\!\!&K&\!\!\! ({\bf r},{\bf r};-i\hbar\beta)\approx
    {L^2\, m\omega\over 2\pi\hbar\,\mbox{senh}(\hbar\beta\omega)}\nonumber\\
    \nonumber\\
    &=& {L^2\,  eB\over 2\pi\hbar\left(\mbox{\large $
    e^{\hbar\beta\omega}$}-
    \mbox{\large $ e^{-\hbar\beta\omega}$}\right)}\nonumber\\
    \nonumber\\
    &=& {L^2\,  eB\over 2\pi\hbar} {\mbox{\large $ e^{-{1\over
    2}\hbar\beta\omega_c}$}\over \left(1-\mbox{\large $
    e^{-\hbar\beta\omega_c}$}\right)}
    \nonumber\\
    \nonumber\\
    &=& \sum_{n=0}^\infty {L^2\,  eB\over 2\pi\hbar} \mbox{\large $
    e^{-\beta(n+{1\over2})\hbar\omega_c}$}\; ,\nonumber\end{aligned}$$ where we denoted by $\omega_c = eB/2m$ the ciclotron frequency. Comparing this result with that of a partition function whose energy level $E_n$ has degeneracy $g_n$, given by $$Z(\beta) = \sum_n g_n\; \mbox{\large $ e^{-\beta E_n}$}\; ,$$ we imediately identify the so called Landau leves and the corresponding degeneracy per unit area, $$\label{niveisdeLandau}
    E_n = \left( n+{1\over 2}\right)\hbar\omega_c\;\; ;\;\;
     