Absence of mobility edges in mosaic Wannier-Stark lattices

Mobility edges, separating localized from extended states, are known to arise in the single-particle energy spectrum of certain one-dimensional models with quasiperiodic disorder. Recently, some works claimed rather unexpectedly that mobility edges can exist even in disorder-free one-dimensional models, suggesting as an example the so-called mosaic Wannier-Stark lattice where a Stark potential is applied on every M site of the lattice. Here, we present an exact spectral analysis of the mosaic Wannier-Stark Hamiltonian and prove that, strictly speaking, there are not mobility edges, separating extended and localized states. Specifically, we prove that the energy spectrum is almost pure point with all the wave functions displaying a higher than exponential localization with the exception of (M - 1)-isolated extended states at energies around which a countably infinite number of localized states with a diverging localization size condense.