Continuum-limit broadening of the dispersion relation

Prove or refute that, in the continuum-wavenumber limit (Δk→0) of the Schrödinger–Helmholtz equation, the discrete decorations of the linear dispersion relation observed in finite-size simulations become a general broadening of the dispersion relation across all wavenumbers.

Background

Finite-resolution simulations (Δk=1) show discrete scarring or decoration along the wave dispersion relation in the (k,ω) spectrum, which the authors interpret as finite-size artifacts. They conjecture that in physical systems with continuous wavenumbers, these decorations would manifest as a broadening of the dispersion relation.

Establishing this would connect numerical signatures to physical continuum behavior and help quantify bound-state–wave interactions in nonlocal media.

References

We conjecture that in a physical system, the result of this decoration at every continuous wavenumber would be a general broadening of the dispersion relation.

— A bound state attractor in optical turbulence (2410.12507 - Colleaux et al., 16 Oct 2024) in Section 3.2 (Emergence and consolidation of a dominant bound state)