I have a problem which I suspect appears in literature under a name I haven't found yet.
Let $H:\ell^2(\mathbb{Z}^2)\to \ell^2(\mathbb{Z}^2)$ given by $H=\Delta + D$, where $\Delta$ is the graph Laplacian on $\mathbb{Z}^2$ and $D$ is a diagonal operator $D \cdot \delta_{(m_1,m_2)}= V(m_1,m_2)\cdot\delta_{(m_1,m_2)}$ such that $V:\mathbb{Z}^2\to[0,\infty)$ attains finitely many values. I consider the projection of $H$, from $\ell^2(\mathbb{Z}^2)$ to the space of functions supported on $[-N,N]^2\cap \mathbb{Z}^2$, and denote it by $H_N.$
Since the off diagonal entries in each row sum to at most $4$, the Gershgorin disc theorem tells me that the eigenvalues of $H_N$ cluster in discs of radius $4$ around the values $V$ takes on $[-N,N]^2\cap \mathbb{Z}^2$. Moreover, each disc contains as many eigenvalues as the corresponding value $V$ is obtained in $[-N,N]^2\cap \mathbb{Z}^2$.
I was wondering whether one can generate a similar finer cover. Is there a known way to build an isospectral operator $\tilde H= \Delta +\tilde D$ such that $\tilde D$ is a diagonal operator assigning some value to each block of the form $$\begin{pmatrix} V(m_1-1,m_2+1) & V(m_1,m_2+1) & V(m_1+1,m_2+1)\\ V(m_1-1,m_2) & V(m_1,m_2) & V(m_1+1,m_2) \\ V(m_1-1,m_2-1) & V(m_1,m_2-1) & V(m_1+1,m_2-1) \end{pmatrix}$$
and then apply a Gershgorin disc theorem for the operator $\tilde H_N$?
