Chiral phase dependence of fermion partition function in two dimension

The chiral phase dependence of fermion partition function in spherically symmetric U(1) gauge field background is analyzed in two dimensional space-time. A well-defined method to calculated the path integral which apply to the continuous fermion spectrum is described. The one-to-one correspondence between the nonzero energy continuous spectra of two pertinent hamiltonians, which are defined by the Dirac operator to make the path integral well-defined, is shown to be exact. The asymptotic expansion for the chiral phase dependence in $1/|m|^2$ ($m$ is mass of the fermion.) is proposed and the coefficients in the expansion are evaluated up to the next-to-next leading term. Up to this order, the chiral phase dependence is given only by the winding number of background field and the corrections vanish.