I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-lattice, and $V$ is a diagonal operator of the form $[ V\psi ](n,m)=u(n,m)\cdot \psi(n,m)$ for any function $\psi\in \ell^2(\mathbb{Z}^2)$ and $(n,m)\in \mathbb{Z}^2$, where $u:\mathbb{Z}^2\to \mathbb{R}$ is some function.

Concretely, I am trying to compare $\sigma_{ess}(H_1)$ and $\sigma_{ess}(H_2)$, where $u_1\equiv 0$ and

$$ u_2(m_1,m_2)= \begin{cases} 16 &;m_1=0, m_2\geq 0 \\ 0 &;\text{else} \end{cases} \; . $$

It is easy to show that $\Vert H_1 \Vert= 4$ and $ \sqrt{20} \leq \Vert H_2\Vert\leq 20$. Since the operators are self-adjoint, we know that $\lambda'\in \sigma(H_2)\setminus \sigma(H_1)$ for $\lambda':= \Vert H_2\Vert $. I am now trying to show that there exists $\lambda \in \sigma_{ess}(H_2)\setminus \sigma_{ess}(H_1)$ with $\lambda\geq 12$, hopefully $\lambda'=\lambda$ itself works. I want to find a sequence of normalized functions $\psi_n\in \ell^2(\mathbb{Z}^2)$ such that $\Vert (H_2-\lambda I)\psi_n\Vert \to 0$ and $\psi_n\to 0$ weakly. It seems like the spectral projection onto $\chi_{[12,20]}(H_2)$ should be infinite dimensional, but I can't manage to show this directly. I thought to look at increasing finite sections around the origin which would each have a strictly growing dimension of projection onto $[12,20]$ using a Gershgorin circle argument, but this hasn't worked so far.

It feels like there should be a simple argument I am missing that someone well versed in spectral theory would notice. I would appreciate any input on this question.