Extended supersymmetry and its reduction on a circle with point singularities

We investigate N-extended supersymmetry in one-dimensional quantum mechanics on a circle with point singularities. For any integer n, N = 2n + 1 supercharges are explicitly constructed. in terms of discrete transformations, and a class of singularities compatible with supersymmetry is clarified. In our formulation, the supersyrnmetry can be reduced to M-extended supersymmetry for any integer M < N. The degeneracy of the spectrum and spontaneous supersymmetry breaking are also studied.