  In this letter the fractional fermion number of thick domain walls is
computed. The analysis is achieved by developing the heat kernel expansion of
the spectral eta functon of the Dirac Hamiltonian governing the fermionic
fluctuations around the domain wall. A formula is derived showing that a non
null fermion number is always accompanied by a Hall conductivity induced on the
wall. In the limit of thin and impenetrable walls the chiral bag boundary
conditions arise, and the Hall conductivity is computed for this case as well.
