  The decomposition of $Spin^{c}(4)$ gauge potential in terms of the Dirac 4%
-spinor is investigated, where an important characterizing equation $\Delta
A_{\mu}=-\lambda A_{\mu}$ has been discovered. Here $\lambda $ is the vacuum
expectation value of the spinor field, $\lambda =\Vert \Phi \Vert ^{2}$, and
$A_{\mu}$ the twisting U(1) potential. It is found that when $\lambda $ takes
constant values, the characterizing equation becomes an eigenvalue problem of
the Laplacian operator. It provides a revenue to determine the modulus of the
spinor field by using the Laplacian spectral theory. The above study could be
useful in determining the spinor field and twisting potential in the
Seiberg-Witten equations. Moreover, topological characteristic numbers of
instantons in the self-dual sub-space are also discussed.
