Question

Some suggestions:

1. To show there is a spectral gap, one can use the following approach: The variational characterization of the first eigenvalue $$ E_1 = \inf_{\|\psi\| = 1} \langle \psi, H \psi \rangle $$ can be used to obtain an upper bound on the first eigenvalue. Then Temple's inequality (I think it's in Reed-Simon IV) can be used to obtain a lower bound on the second eigenvalue. Note in order to show the existence of a spectral gap, one needs to study all the operators $H(k)$ of the Floquet-Bloch decomposition.

2. But I would guess that at least as long as $h$ is "sufficiently small", there is no spectral gap (without having done any of the computations). Denote by $E_1(k)$ and $E_2(k)$ the first and second eigenvalue of $H_0(k)$. Here $H_0$ denotes the usual Laplacian and the Floquet--Bloch decomposition is done with respect to $[0,1]^2$. I believe that one has that $$ \sup_{k} E_1(k) > \inf_{k} E_2(k). $$ Then using that your operator will be a small perturbation of the Laplacian (in an appropriate sense), one should obtain that there is no spectral gap.

   2b. (added in edit) It is clear that for fixed $k$, one has $E_1(k) < E_2(k)$. This follows from the first eigenfunction being positive.

3. There is what is called "Bohr-Sommerfeld conjecture", which is related to higher eigenvalues.