The complexity of LH with constant gap
Your original problem formulation is not correct. It's missing some normalization on $H$, otherwise it could be the case that all eigenvalues of $H$ are always smaller than $a$, making the problem completely trivial, so it couldn't possibly be QMA-complete.
One natural normalization is $||H||\leq 1$, which you can enforce by saying that if $H$ is expressed as the sum of $m$ local terms, then each term should have norm at most $1/m$. Now all eigenvalues of $H$ must lie in $[-1,1]$. For this normalization, the complexity of constant-gap Local Hamiltonian is conjectured to be QMA-hard. Note that this problem is NP-hard because of the classical PCP theorem.