7 problems

Let $H$ be a periodic elliptic second-order operator with sufficiently smooth coefficients. Kuchment's conjecture. The operator $H$ does not admit any flat bands. This conjecture i…

Nonexistence conjecture. The flat bands under consideration do not have compact localized states.

Nonexistence conjecture. Unlike flat bands in short-ranged translationally invariant Hamiltonians, the considered $E=0$ flat bands do not host compact localized states.

Korotyaev–Saburova's conjecture. The perturbed operator $\mathcal{A}_\Gamma+Q$ should have no flat bands.

No-flat-band conjecture. The maximal abelian cover of $G$ should have no flat bands.

Let a Hubbard model be obtained by adding a small perturbation to the hopping Hamiltonian of a flat-band model, so that its lowest single-electron band is slightly dispersive, and…

Mielke–Tasaki stability-class conjecture. Mielke–Tasaki flat-band ferromagnetism is related to a new stability class.