3 problems

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.

Let a graph carry a Hamiltonian whose eigenstates include localized states, and consider the return probability of a quantum walk to its starting point. High-energy localized-state…