Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several talks about a quasimode construction due to Ralston (and maybe Babich?), but don't know of a precise reference. Here is what I loosely understand. Given any natural number $k$ (where usually $k \to \infty$), one can define a function $\psi_k$ (called a "quasimode") satisfying the following properties:
$\psi_k$ is supported in a tubular neighbourhood $T$ around $\gamma$ of radius $\sim k^{-{1/2}}$
$\psi_k$ satisfies that $\left(-\Delta - \frac{4\pi^2k^2}{L^2}\right)\psi_k$ is small in $L^2$-sense, and goes to $0$ (or at least bounded) as $k \to \infty$. $\psi_k$ looks like $e^{ik\psi}\phi(t, \overline{x})$, where we are using Fermi coordinates around the geodesic $\gamma$, and $\psi, \phi$ should satisfy reasonable pdes (I think $\psi$ solves some eikonal equation).
Is my understanding above correct? Also, I would be grateful if someone points out to some references where the above construction appears.
