energy gap from the free states; therefore, it is natural to expect that at $T=0$ the system should superconduct. This result is already a well established fact on the basis of Hartree-Fock analysis[@6] and Random Phase Approximation [@6b],[@10a]. The case at $T\neq 0$ is more controversial since thermal fluctuations, occurring in the many-particle system, can produce significant changes. As we will show in this paper, the presence in this theory of a natural scale, the cyclotron frequency $\omega _{c}$, is crucial for the existence of a phase at $T\ll \omega _{c}$, on which the system, when confined to a bounded region, still behaves as a superconductor.

The fermion thermal Green’s function in the presence of the background Chern-Simons field $\overline{b}$

$$G\left( x,x^{\prime }\right) =-\left\langle T_{\tau }\psi \left( x\right) 
\overline{\psi }\left( x^{\prime }\right) \right\rangle  \eqnum{2.13}$$

is obtained by solving the equation

$$\left( \partial _{\tau }-\frac{1}{2m}\overline{D}_{k}^{2}-\mu \right)
G\left( x,x^{\prime }\right) =-\delta _{3}\left( x-x^{\prime }\right) 
\eqnum{2.14}$$

subject to the requirement of antiperiodicity under the imaginary time translation $\tau \rightarrow \tau +\beta $ ($\beta $ is the inverse absolute temperature). In (2.14) we have introduced the notation

$$\overline{D}_{k}=\partial _{k}+i\overline{a}_{k}  \eqnum{2.15}$$

The Fourier transform of the fermion thermal Green’s function (2.13)

$$G\left( p_{4},{\bf p}\right) =\int\limits_{0}^{\beta 