Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representation

We study the nonlinear (polynomial, N-fold,...) supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a nonlinear superalgebra which can be appropriately specified as N = 2 SUSY. In this case we find a nontrivial hidden symmetry operator and rephrase it as a nonlinear function of the Hamiltonian on the physical state space. The full N = 2 nonlinear SUSY algebra includes "central charges" both polynomial and nonpolynomial (due to a symmetry operator) in the Hamiltonian. (C) 2003 Elsevier Science B.V. All rights reserved.