Some question about the spectral function of Laplace operator on $\mathbb{R}^n$

Question

I am studying the paper of Seeley, R., A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of (R^3), Adv. Math. 29, 244-269 (1978). ZBL0382.35043. There are some question about the spectral function $e(x,y,\lambda)$ confused me very much, I hope someone can help me.

Consider the Laplace operator $\Delta=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ in $\mathbb{R}^n$, and let $E_{\lambda}$ be the spectral resolution of $\Delta$. I know that $E_{\lambda}$ is a projector from $L^{2}(\mathbb{R}^n)$ to $L^{2}(\mathbb{R}^n)$ for any $\lambda\in \mathbb{R}$. By using of the Schwarz kernel Theorem, we can obtain a Schwarz kernel $e(x,y,\lambda)$ of $E_{\lambda}$, and $e(x,y,\lambda)\in \mathcal{D}'(\mathbb{R}^n\times\mathbb{R}^n)$ for any $\lambda>0$. It seems by the regularity estimates of Laplace operator, we can conclude that $e(x,y,\lambda)\in C^{\infty}(\mathbb{R}^n\times\mathbb{R}^n)$ for any fixed $\lambda>0$. In many papers (e.g. [1]), the authors used notation $$ u(x,y,t)=\int_{0}^{\infty}\cos{\lambda t}~d_{\lambda}e(x,y,\lambda^{2}) $$ to the wave kernel of $\Delta$ in $\mathbb{R}^n$, which satisfies $$\left\{ \begin{array}{ll} u_{tt}-\Delta u=0 , & \\ u|_{t=0}=\delta(x-y),\qquad u|_{t=0}=0 & \end{array} \right.$$

For $n=3$, Seeley obtained that $$u(x,y,t)=(2\pi)^{-3}4\pi\int \cos(t\tau) \tau^{2} d\tau=\frac{1}{6\pi^{2}}\int_{0}^{\infty}\cos(t\tau)d\tau^{3} $$ Then, he claimed that the spectral function $e(x,y,\lambda)$ in $\mathbb{R}^3$ is $$ e(x,x,\tau^2)=\frac{1}{6\pi^{2}}\tau^{3} $$ by compare the two equations above.

My questions are as follows:

1. I feel very confused about how can we write the notation $d_{\lambda}e(x,y,\lambda)$? Is $e(x,y,\lambda)$ a function of bounded variation with respect to variable $\lambda$ for fixed $x,y$ that make $d_{\lambda}e(x,y,\lambda)$ as a measure? What's the properties of $e(x,y,\lambda)$ ?

2. How can we recover the spectral function $e(x,x,\tau^2)=\frac{1}{6\pi^{2}}\tau^{3}$ by only compare $$u(x,y,t)=(2\pi)^{-3}4\pi\int \cos(t\tau) \tau^{2} d\tau=\frac{1}{6\pi^{2}}\int_{0}^{\infty}\cos(t\tau)d\tau^{3} $$ with $$ u(x,y,t)=\int_{0}^{\infty}\cos{\lambda t}~d_{\lambda}e(x,y,\lambda^{2})? $$ The first integral does not converge and $u(x,y,t)$ is only a distribution.

I find many books but get nothing, can someone give some reference about the fundamental theory of spectral function and wave kernel for elliptic operators? Thank you very much!

Reference [1] Ivrii, Victor, 100 years of Weyl’s law, Bull. Math. Sci. 6, No. 3, 379-452 (2016). ZBL1358.35075.

Answer 1

This integral is understood as a Fourier transform of a temperate distribution and it is itself a temperate distribution.

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From comments below:

1: How to understand $\int f(\tau) d\tau e(x,x,\tau)$: Since for Laplacian and elliptic operators in general $e(x,x,\tau)$ is a smooth function of $x, y$ with a value in the space of temperate distributions, there is no reason to worry how to understand this integral. Furthermore, it is not a measure, but more singular distribution in higher dimensions.

2: We do not recover $e(x,x,\tau)$ from $u(x,x,t)$ for all $t$ as in most cases it is impossible to construct an approximation for all $t$, but only for $t$ in small vicinity of $0$. Then we use Tauberian theorem of Hormander. I suggest to start from more elementary books; M. Shubin, Pseudodifferential operators and spectral theory would be the best.