Superconformal mechanics and nonlinear supersymmetry

We show that a simple change of the classical boson-fermion coupling constant, 2α → 2αn, n ∈ ℕ, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) × u(1) subalgebra, and by the 2(n + 1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter α is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with |α| = p, p ∈ ℕ, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry.