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!