Quantum States far from the Energy Eigenstates of Any Local Hamiltonian

What quantum states are possible energy eigenstates of a many-body Hamiltonian? Suppose the Hamiltonian is nontrivial, i.e., not a multiple of the identity, and L local, in the sense of containing interaction terms involving at most L bodies, for some fixed L. We construct quantum states ψ which are ``far away'' from all the eigenstates E of any nontrivial L-local Hamiltonian, in the sense that ψ-E is greater than some constant lower bound, independent of the form of the Hamiltonian.