Why do people study Weyl asymptotics and partial-spectral-projections? The major focus of the research that my advisor has me doing centers around the idea of asymptotic behavior of partial-spectral-projections on compact manifolds. In a few sentences, here is the context for the research:

  
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*$(M,g)$ is a compact Riemannian manifold without boundary, and $-\Delta_g$ is the (positive) Laplace-Beltrami operator of the metric $g$.
  
*The operator $\sqrt{-\Delta_g}$ is defined in the usual way its collection of $L^2$-normalized eigenfunctions is denoted by $\{e_j(x)\}_{j=0}^{\infty}$, with eigenvalues $0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \cdots \to \infty$.
  
*For a fixed $\lambda > 0$, we then define the partial-projection operator $$ \sum_{j=0}^{\infty}\langle f,e_j\rangle e_j(x) = f(x) \mapsto \sum_{\{j \,:\, \lambda_j \in [\lambda, \lambda+1)\}} \langle f,e_j \rangle e_j(x) $$ as the projection of $f(x)$ onto the direct sum of eigenspaces which have eigenvalues in the unit-interval $[\lambda, \lambda+1)$.
  
*We then denote the Schwartz kernel of the corresponding integral operator as $K(x,y;\lambda)$, where $$ f(x) \mapsto \int_{M} K(x,y;\lambda)f(y) \,dV_g(y) $$  agrees with the partial-sum definition above.
  
*The goal of our research is to then analyze the big-oh behavior of this Schwartz-kernel as $\lambda \to \infty$. Usually this is formulated as $$ \sup_{x,y \in M}\big| K(x,y;\lambda) - F(x,y;\lambda) \big| = O(\lambda^{n-1}), $$ where the term $F(x,y;\lambda)$ comes from some parametrix approximation or something.

At this point I'm a little embarrassed to say admit that while I can do the mathematical research needed, I am unsure as to why people actually care about such a specific kind of linear operator?
I understand that the Weyl law is an old result in functional and harmonic analysis, but sadly I'm not sure why this specific problem is useful in the larger field of research. I've tried asking this of my advisor before, but he has not offered me all that much in the way of an answer. Also, while reading through the literature of similar problems to my own, I find many references to the myriad of results and slightly-different hypotheses, but still an answer of WHY? eludes me.
Specifically, why does everyone also study these partial-projections onto unit-length interval? What would be different if we projected only an interval of length 2? Or length $L$? Or onto a compact set of some fixed, finite measure?
Any insight into these kinds of problems, and their important to the mathematical body at-large would be much appreciated. Thank you in advanced, as always.
 A: The unit length hypothesis at here is not important, and very crude estimates are available using Sobolev embedding only. The main issue is that studying the spectrum on the manifold itself is not enough for recovering underlying topological/geometric information of the manifold. This is a subtle topic even for 2 dimensional surfaces, where a lot of work has been done. 
For very recent work, check some papers by Sogge and Xi:
https://arxiv.org/abs/1711.04707
I would suggest that instead of working through the detailed estimates (on a sphere, on a torus, on negatively curved manifold, etc), think about some other ways to understand the spectrum of the Laplacian on the manifold. For example, a compact Riemann surface of genus $g\ge 2$ can be realized as the quotient of the upper half plane $\mathcal{H}/\Gamma$. There is a lot of interesting work that can be done to understand the relationship between the group action and the spectrum. The interplay between the algebraic nature of the surfaces and flexibility of the analysis tools made the subject really interesting. 
A survey paper by Sanark may be a good start:
http://web.math.princeton.edu/facultypapers/sarnak/baltimore.pdf
For 3-manifolds this becomes deep and is related to heat kernels in geometric analysis. The subject is related to Ricci flow and there are plenty written up online already. 
