I'm reading some material on mathematical physics where the so-called discrete (or lattice) Laplacian takes place. Let me fix some notation. Consider a fixed (finite) $\Lambda \subset \mathbb{Z}^{d}$ and vectors $\varphi = (\varphi_{x})_{x\in \Lambda}\in \mathbb{R}^{\Lambda}$. Furthermore, the set $\{\hat{e}_{1},...,\hat{e}_{d}\}$ is the canonical ordered basis for $\mathbb{Z}^{d}$, i.e $e_{j}$ is a vector in $\mathbb{Z}^{d}$ with all entries equal to zero except for the $j$-th one, which is equal to one. Analogously, $\{e_{1},...,e_{\Lambda}\}$ forms a basis for $\mathbb{R}^{\Lambda}$ in the same fashion. For each $j=1,...,d$, set $\nabla_{j}\varphi(x) := \varphi(x+\hat{e}_{j}) - \varphi(x)$ and: \begin{eqnarray} (\nabla\varphi)^{2}(x) := \sum_{j=1}^{d}(\nabla_{j}\varphi(x))^{2} \tag{1}\label{1} \end{eqnarray} Now, the discrete Laplacian $\Delta$ is defined to be the unique linear operator satisfying: \begin{eqnarray} \langle \varphi, -\Delta\varphi\rangle = \sum_{x\in \Lambda}(\nabla \varphi)^{2}(x) \tag{2}\label{2} \end{eqnarray} for every $\varphi \in \mathbb{R}^{\Lambda}$, where $\langle \varphi, \psi\rangle := \sum_{x\in \Lambda}\varphi_{x}\psi_{x}$.
First Question: The uniqueness of $\Delta$ in Definition (\ref{2}) is easy to see, but the existence does not seem obvious. How can I prove the existence part?
Second Question: By (\ref{2}), $-\Delta$ is positive-definite, so that it has an inverse $(\Delta-m^{2})^{-1}$, for every $m \in \mathbb{R}$. How can I obtain this inverse operator? It seems tha I should obtain its discrete Green's function, but how to define such an object? Is it demanding that $(-\Delta+m^{2}) G(x-y) = \delta_{xy}$ where $\delta_{xy}$ is the Kronecker delta?
Third Question: If I set $\nabla \varphi(x) := (\nabla_{1}\varphi(x),...,\nabla_{d}\varphi(x))$ and $\nabla \varphi := \sum_{x\in \Lambda}e_{x}\otimes \nabla \varphi(x)$, where $\otimes$ denotes the usual Kronecker product of matrices, we should have: \begin{eqnarray} \nabla\varphi^{T}\nabla\varphi = \sum_{x\in \Lambda}\sum_{y\in \Lambda}e_{x}^{T}e_{y}\otimes \nabla\varphi(x)^{T}\nabla\varphi(y) = \sum_{x\in \Lambda}(\nabla\varphi)^{2}(x) \equiv \langle \varphi, -\Delta \varphi\rangle \end{eqnarray} So, this seems to be a more rigorous (or at least more structured) way to define and give meaning to the symbols presented in (\ref{1}) and (\ref{2}). Is this correct?