this question is somewhat naive, but I am trying to understand the meaning of the regularized resolvant of the Laplacian on $\mathbb{R}^d$, and how it relates to a discrete approximation. Particularly, I am in the situation where have a formal definition given to me by a physics collegue, that I want to understand.
So my understanding is that one may formally define a version of the resolvant for the Laplacian of $\Delta_d$ on $\mathbb{R}^d$ as $$\mathrm{tr}(\mu+\Delta_d)^{-1}=\frac{1}{\pi^d}\int_{\mathbb{R}^d}\frac{d\omega}{\mu+\|\omega\|^2}.$$
When $d=1$, this is convient, as you just get $$\frac{1}{\pi}\int_{\mathbb{R}}\frac{d\omega}{\mu+\omega^2}=\frac{1}{\sqrt{\mu}}.$$
For $d>1$, you have a problem, as this integral is diveregent. However, you can fix this for $d=2,3$ by instead considering the difference: $$\mathrm{tr}(\mu+\Delta_d)^{-1}-\mathrm{tr}(\mu_0+\Delta_d)^{-1}.$$
For $d>3$, instead consider the derivatives $$\mathrm{tr}(\mu+\Delta_d)^{-j}=\frac{1}{\pi^d}\int_{\mathbb{R}^d}\frac{d\omega}{(\mu+\|\omega\|^2)^{-j}},$$ and then integrate the answer as appropriate.
My question is what this means, and how it relates to finite settings, where the resolvant is well defined. My combinatorics upbringing lead me to try to approximate the Laplacian on $\mathbb{R}^d$ by the rescaling the Laplacian, $\Delta_{d,N}$ on the discrete graph on $\mathbb{Z}^d\cap [1,N]^d$, considered as a periodic graph. This is nice as it has a finite set of explicit eigenvalues. I can show, for example, if you denote the $\Delta_{1,N}$, and take an $N$-dependent spacing $\\epsilon_N=N^{-1/2}$, then $$\lim_{N\to \infty}\mathrm{tr}(\mu \epsilon_N^{-1/2}-\Delta_{1,N}\epsilon_N^{1/2})^{-1}=\frac{1}{\sqrt{\mu}}.$$
However, I am not sure how to show this for $d=2,3$, or even what the meaning of these formal quantities are. Looking for references, I have found a number of results for such approximations for compact Riemannian manifolds, but as the continuum resolvant there is already defined, I haven't found anything related to this. In particular, a reference or explanation for why $d=4$ is critical in terms of the actual behavior of the resolvant, or a discrete approximation would be greatly appreciate
