Aharonov-Bohm problem on a square lattice

An explicit expression is obtained for the Green's function of the Schrödinger operator describing the motion of an electron on a square lattice in a magnetic field with flux that pierces a single cell it is show that the nontrivial component of the Green's function, equal to the contribution from the paths that pass round the singularity, is a compact operator. The method can also be used to find the partition functions for the ensembles of paths on the lattice that are associated with the numbers of turns around the fixed point.