  We propose that in noncentrosymmetric superconductors with weakly asymmetric
spin-orbit interaction the field-induced pair correlation between the
spin-orbit split different bands ignored in previous studies yields unique
effects; i.e. the Pauli depairing effect is anisotropic in the momentum space,
and as a result, magnetic fields induce point-node-like anisotropic gap
structure of the quasiparticle energy even for isotropic s-wave states, which
seriously affects thermodynamic quantities at low temperatures. Also, it is
shown that when the magnitude of the spin-orbit interaction is smaller than the
superconducting gap, the specific heat as a function of a magnetic field
exhibit a two-gap-like behavior, even when there is only a single gap. These
features characterize parity violated Cooper pairs in weakly noncentrosymmetric
systems. We suggest the possible detection of these effects in the
superconductor with weakly broken inversion symmetry Y_2C_3.
