Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on [0,1]^2.


I'm trying to estimate the eigenfunctions and eigenvalues of the Laplace-Beltrami operator for this surface. Of course, the laplace beltrami operator can be very easily be expressed in local coordinates on $[0,1]^2$, and so the problem is effectively approximating the spectra of this second order elliptic operator on $L^2[0,1]^2$ with periodic boundary conditions. This is what I am doing, however I still cannot obtain any useful estimates.


Is anyone aware of any references, or has had any experience with this problem?



Estimating laplace-beltrami spectra for a graph surface in $R^3$
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.


