Me being a mathematician, I tend to do the usual “the question you’re asking may not be the question you should ask”. I’m sure that you know a lot of people with the same professional deformity of character.
I get that this can be aggravating to some people. Instead of getting angry, you should just give us more context on what you need, and why you believe you need it. The entire discussion could have been less acrimonous if you had started off with e.g. "hey guys, I know this question might seem weird but we need the full spectrum of a so-and-so operator coming from a so-and-so PDE. This is because … [e.g. hyptothetical: … because we don’t want to study the PDE, we instead want to study the behavior of this specific discretization that many people use, especially to study how the distortions that are introduced at wave-lengths close to the grid size impact higher-level results]. "
And you should be open-minded to the possibility that there are shortcuts that you did not think about, and that these might be hiding in the gaps where your question is (intentionally?) vague and missing context.
Can you elaborate what “knowledge of the full spectrum” means for your purposes? What is “a spectrum” for you and what is the notion of accuracy / approximation that you need? Can you formulate this in a way that embeds all different N, preferably in a way such that N->\infty permits finite-support approximations?
To give a hypothetical style of answer that would open the door to clever tricks:
Ok, I have an accuracy \epsilon that comes from the discretization error (no computation needs to be better than the discretization error we incurred anyways!). We formulated the problem such that the spectrum is uniformly bounded away from \infty. We don’t care about dimensions, we just want to know where the spectrum is. That means: We want to identify all points with distance > epsilon from an eigenvalue of the matrix, and we want to identify all points with distance < epsilon/2 from an eigenvalue; for points with epsilon/2 < dist < epsilon we don’t care (i.e. these may be misidentified and this is where numerical inaccuracies and your clever algorithms should come in!)". Here is a plot for 3 small values of N, so what you can see some qualitative behavior.