This dissertation consists of two parts. In the first half, we construct a frame of complex Gaussians for the space of $L^2(\mathbb{R}^n)$ functions. When propagated along bicharacteristics for the wave equation, the frame can be used to build a parametrix with suitable error terms. When the coefficients of the wave equation have more regularity, propagated frame functions become Gaussian beams.

In the latter half, we consider two dimensional real-valued analytic potentials for the Schroedinger equation which are periodic over a lattice $\mathbb{L}$. Under certain assumptions on the form of the potential and the lattice $\mathbb{L}$, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of $C^{\infty}(\mathbb{R}^2/\mathbb{L})$ potentials. The result extends the work of Eskin et. al, in "On isospectral periodic potentials in $\mathbb{R}^n$, II."