  We study a pairing mechanism for the quantum Hall system using a mean field
theory with a basis on the von Neumann lattice, on which the magnetic
translations commute. In the Hartree-Fock-Bogoliubov approximation, we solve
the gap equation for spin-polarized electrons at the half-filled Landau levels.
We obtain an effective Hamiltonian which shows a continuous transition from the
compressible striped state to the paired state. Furthermore, a crossover occurs
in the pairing phase. The energy spectrum and energy gap of the quasiparticle
in the paired state is calculated numerically at the half-filled second Landau
level.
