The fermion determinant, its modulus and phase

We consider path integration of a fermionic oscillator with a one-parameter family of boundary conditions with respect to the time coordinate. The dependence of the fermion determinant on these boundary conditions is derived in a closed form with the help of the self-adjoint extension of differential operators. The result reveals its crucial dependence on them, contrary to the conventional understanding that this dependence becomes negligible over sufficiently long time evolution. An example in which such dependence plays a significant role is discussed in a model of supersymmetric quantum mechanics.