Quasiclassical approach to the weak levitation of extended states in the quantum Hall effect

The two-dimensional motion of a charged particle in a random potential and a transverse magnetic field is believed to be delocalized only at discrete energies $E_N$. In strong fields there is a small positive deviation of $E_N$ from the center of the $N$th Landau level, which is referred to as the ``weak levitation'' of the extended state. I calculate the size of the weak levitation effect for the case of a smooth random potential re-deriving earlier results of Haldane and Yang [PRL 78, 298 (1997)] and extending their approach to lower magnetic fields. I find that as the magnetic field decreases, this effect remains weak down to the lowest field $B_{min}$ where such a quasiclassical approach is still justified. Moreover, in the immediate vicinity of $B_{min}$ the weak levitation becomes additionally suppressed. This indicates that the ``strong levitation'' expected at yet even lower magnetic fields must be of a completely different origin.