Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{l=-\lfloor k^{0.99}\rfloor}^{\lfloor k^{0.99}\rfloor}e^{ikx}e^{ily}$
($2\lfloor k^{0.99}\rfloor+1$ is just the number of summands.)
$u_k$, $k\in\mathbb{N}$, is a poor quasimode for the Laplacian:
$||(\Delta-k^2)u_k||_{L^2(\mathbb{T}^2)}=O(k^{1.98})$
(here $\Delta=-(\partial_x^2+\partial_y^2)$).
Via the Dirichlet kernel, we know that
$u_k=\frac{e^{ikx}}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\frac{\sin((\lfloor k^{0.99}\rfloor+\frac{1}{2})y)}{\sin(\frac{y}{2})}$
My question is about $||u_k||_{L^2(\Omega)}$, where $\Omega$ is any region localized away from the closed orbit $\gamma:=\{y=0,x\in S^1\}$ i.e. $\Omega\cup\gamma=\emptyset$. How would I find the value of $\rho$ such that
$||u_k||_{L^2(\Omega)}\leq Ck^{-\rho}$?
I suspect that $\rho>0$ is non-trivial. Any ideas how I can compute the value of $\rho$?
Thanks!
