Our many-body theory for fractional quantized Hall effect is generalized to include $v=\frac{q}{p}$ filling. As the electrons of a two-dimensional system under a strong magnetic field are in a regular array of the Landau orbitals, correlation energy is enhanced and energy gaps are formed. Such a state can give rise to the Hall steps at fractional multiples of $\frac{e^2}{h}$. The motion equation of one-particle Green function provides a systematic theory for calculating energy gaps.

• Received 21 June 1983

DOI:https://doi.org/10.1103/PhysRevB.29.636