Department of Mathematical Physics, Institute of Physics,
Sankt-Petersburg State University,
1 Ulianovskaya Str.,
Petrodvorets, Sankt-Petersburg, 198504, RUSSIA
INFN, Sezione di Bologna,
via Irnerio 46, 40126, Bologna, ITALIA

Nonlinear supersymmetry in quantum mechanics

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 Super-Hamiltonian projection on the zero-mode subspace of a supercharge. We show that the SUSY algebra with transposition symmetry is always polynomial in the Super-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. In the case with two independent supercharges (SUSY algebra in this case can be appropriately specified as N = 2 SUSY) we find a non-trivial hidden symmetry operator and rephrase it as a non-polynomial function of the Super-Hamiltonian on the physical state space. The full N = 2 Nonlinear SUSY algebra includes "central charges" both polynomial and non-polynomial (due to a symmetry operator) in the Super-Hamiltonian. It is shown that wave functions of all Super-Hamiltonian bound states are zero-modes of the hidden symmetry operator.