Schwartz kernel of spectral projection of Laplacian and integrated density of states

Question

I'm reposting here a question I asked on MSE which did not receive an answer.

I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later we will change that.

For $E\in\mathbb R$, I can use the functional calculus (or spectral theorem) to define the operator $\chi_{(-\infty, E]}(\Delta)$, which basically corresponds to projecting onto all eigenspaces up to energy $E$. This operator has an associated Schwartz kernel $S(x,y)$, i.e., an integral kernel:

$$\chi_{(-\infty, E]}(\Delta)f=\int_{y\in U} S(x,y)f(y)dy.$$

I am interested in a relation between $S(x,y)$ and the integrated density of states of $\Delta$. Since the spectrum of $\Delta$ is discrete, the integrated density of states is just the normalized counting function:

$$N(E)=\frac{\#\{\lambda\in \text{spec}(\Delta):\lambda \leq E\}}{|U|}.$$

Using some elementary arguments, I was able to show that

$$S(x,y)=\sum_{\lambda\leq E}g_{\lambda}(x)\overline{g_{\lambda}(y)},$$ where the summation is over all eigenvalues up to $E$ and $g_{\lambda}$ is the noramlized eigenfunction associated to $\lambda$. Using this identity, straightforward integration shows that $$N(E)=\frac{\int_{x\in U} S(x,x)dx}{|U|}.$$

So far so good. But my real interest is actually to study $U$ which is unbounded, and so the spectrum of $\Delta$ is no longer discrete. In the unbounded setting, we define the integrated density of states as

$$N(E)=\lim_{n\rightarrow \infty} \frac{\#\{\lambda\in \text{spec}(\Delta|_{U_n}):\lambda \leq E\}}{|U_n|},$$ where the increasing sequence of subdomains $U_n\subset U$ converges to $U$ "nicely" (don't worry about technicalities, just assume it makes sense in our setting).

In this case, the relation above between $S(x,y),N(E)$ clearly no longer holds, as there are (usually) no eigenfunctions and $|U|$ is infinite, so the expression no longer makes sense.

I believe that one can replace the summation over $g_\lambda$ with integration over spectral projections (with respect to certain spectral measures), but I am not very well versed in these sorts of things. I am also not sure what would replace the normalization factor, as it is infinite now.

Does anyone have any idea on how to prove a similar relation in the infinite setting? Or if it already exists somewhere, I'd also appreciate a reference.

Thanks in advance.

This post imported from StackExchange MathOverflow at 2024-11-02 20:47 (UTC), posted by SE-user GSofer