Abstract

We show that an n-gap periodic quantum system with parity-even smooth potential admits 2n - 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral tri-supersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lamé equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite-period limit. © 2008 IOP Publishing Ltd.