Finite-gap systems, tri-supersymmetry and self-isospectrality

TL;DR

This paper explores the construction of isospectral supersymmetric extensions of finite-gap periodic quantum systems, revealing a tri-supersymmetry structure and conjecturing self-isospectrality for systems with antiperiodic states, supported by explicit elliptic system examples.

Contribution

It introduces a novel tri-supersymmetry framework for finite-gap systems and proposes a conjecture on self-isospectral extensions, supported by explicit elliptic system constructions.

Findings

Existence of $2^n-1$ isospectral super-extensions for n-gap systems

Construction of self-isospectral tri-supersymmetric pairs in elliptic systems

Recovery of broken tri-supersymmetry in the infinite period limit

Abstract

We show that an n-gap periodic quantum system with parity-even smooth potential admits $2^n-1$ 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 Lame 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.