  In the previous work, it was shown that, in supersymmetric (matrix)
discretized quantum mechanics, inclusion of an external field twisting the
boundary condition of fermions enables us to discuss spontaneous breaking of
supersymmetry (SUSY) in the path-integral formalism in a well-defined way. In
the present work, we continue investigating the same systems from the points of
view of localization and Nicolai mapping. The localization is studied by
changing of integration variables in the path integral, which is applicable
whether or not SUSY is explicitly broken. We examine in detail how the
integrand of the partition function with respect to the integral over the
auxiliary field behaves as the auxiliary field vanishes, which clarifies a
mechanism of the localization. In SUSY matrix models, we obtain a matrix-model
generalization of the localization formula. In terms of eigenvalues of matrix
variables, we observe that eigenvalues' dynamics is governed by balance of
attractive force from the localization and repulsive force from the Vandermonde
determinant. The approach of the Nicolai mapping works even in the presence of
the external field. It enables us to compute the partition function of SUSY
matrix models for finite N (N is the rank of matrices) with arbitrary
superpotential at least in the leading nontrivial order of an expansion with
respect to the small external field. We confirm the restoration of SUSY in the
large-N limit of a SUSY matrix model with a double-well scalar potential
observed in the previous work.
