Electron states in a magnetic field and random impurity potential: Use of the theory of entire functions

The authors determine the wavefunction of an electron in the presence of a transverse magnetic field and randomly located delta function impurities in the x-y plane within the subspace of the lowest Landau level. The wavefunction contains as a factor an entire function of z=x+iy which vanishes at all impurity sites. The question of whether such states are extended is related to the rate of growth of an entire function in terms of the distribution of its zeros. For a homogeneous distribution of impurities the corresponding Weierstrass product is of order 2 and of finite type. This rate of growth can be exactly compensated by the Gaussian term due to the presence of the magnetic field such that there is a critical field beyond which extended states may exist. If the impurities are located on the sites of a square lattice, the extended states are given in closed form in terms of the Weierstrass sigma function.