The $U(1)$ anomaly and zero modes in two-dimensional Euclidean quantum electrodynamics

For the quantum fermion field wave function the problem of the boundary condition choice consistent with the nontrivial topology of the external fields is concidered. The boundary conditions on the boundaries of the two-dimensional domains in the form of the disk and of the bounded cylinder are found. These conditions are equivalent to the square integrability of the zero modes in the noncompact domains.