Department of Mathematical Physics, Institute of Physics,

Sankt-Petersburg State University,

1 Ulianovskaya Str.,

Petrodvorets, Sankt-Petersburg, 198504, RUSSIA

E-mail: Postmaster@sokolov.pdmi.ras.ru

and

INFN, Sezione di Bologna,

via Irnerio 46, 40126, Bologna, ITALIA

E-mail: Alexandr.Andrianov@bo.infn.it

**Nonlinear supersymmetry in quantum mechanics**

**Abstract:**

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.