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:

• $(M,g)$ 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 $$ \sup_{x,y \in M}\big| K(x,y;\lambda) - F(x,y) \big| = O(\lambda^{n-1}), $$ where the term $F(x,y)$ 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.