Generalization of Janus quantum numbers beyond the AAH model

Establish whether Janus quantum numbers can be defined for every quasiperiodic Hamiltonian that depends on a single parameter and satisfies the Wigner–von Neumann non-crossing condition.

Background

The paper defines a Janus quantum number as the dual labeling of a quasiperiodic eigenstate by momentum-space and real-space indices in different parameter regimes. In the Aubry–André–Harper (AAH) model, eigenstates are localized in momentum space for modulation strength V<2t and labeled by momentum k, while they are localized in real space for V>2t and labeled by a lattice coordinate r. The eigenstate can be continuously tracked across the localization transition because the energy levels generically do not cross as the single parameter V varies.

The authors conjecture that this dual-labeling structure is not specific to the AAH model but applies to any quasiperiodic Hamiltonian depending on one parameter and satisfying the Wigner–von Neumann condition. Establishing this claim would determine the general scope of Janus quantum numbers within quasiperiodic spectral theory.