ar X iv : q ua nt - p h / 01 06 03 7 v 2 1 9 Ju n 20 01 N - fold Supersymmetry in Quantum Mechanics - General Formalism -

We report general properties of N -fold supersymmetry in one-dimensional quantum mechanics. N -fold supersymmetry is characterized by supercharges which are N -th polynomials of momentum. Relations between the anti-commutator of the supercharges and the Hamiltonian, the spectra, the Witten index, the non-renormalization theorems and the quasi-solvability are examined. We also present further investigation about a particular class of N -fold supersymmetric models which we dubbed type A. Algebraic equations which determine a part of spectra of type A models are presented, and the non-renormalization theorem are generalized. Finally, we present a possible generalization of N -fold supersymmetry in multi-dimensional quantum mechanics. [email protected] [email protected] [email protected] 2 Aoyama, Sato and Tanaka, N -fold Supersymmetry