Flat-band exclusion for periodic elliptic second-order operators

Prove that every periodic elliptic second-order operator with sufficiently smooth coefficients admits no flat bands, thereby resolving Conjecture 5.18 for general periodic elliptic second-order operators.

Background

The paper presents its generic no-flat-band theorem for discrete periodic graph operators as a discrete analogue of Conjecture 5.18 from Kuchment’s survey. The cited conjecture concerns continuous periodic elliptic second-order operators with sufficiently smooth coefficients and asserts that their dispersion relations cannot contain flat bands.

The authors note that the assertion is known for the continuous Schrödinger operator H = Δ + V, but remains unresolved for general periodic elliptic second-order operators. Thus, the open problem lies outside the discrete theorem proved in the paper and is not answered by the paper’s results.

References

A periodic elliptic second-order operator H with sufficiently smooth coefficients does not admit any flat bands. For the continuous Schrödinger operator H = \Delta + V, it is well known that no flat bands occur , yet Conjecture~5.18 remains largely open for general periodic elliptic second-order operators.

— Rare Flat Bands for Periodic Graph Operators  (2503.03632 - Faust et al., 5 Mar 2025) in Section 1, Introduction and main result

For the continuous Schrödinger operator $H = \Delta + V$, it is well known that no flat bands occur , yet Conjecture~5.18 remains largely open for general periodic elliptic second-order operators.

— Rare Flat Bands for Periodic Graph Operators  (2503.03632 - Faust et al., 5 Mar 2025) in Section 1, Introduction