I am studying the paper of <cite authors="Seeley, R.">_Seeley, R._, [**A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of \(R^3\)**](http://dx.doi.org/10.1016/0001-8708(78)90013-0), Adv. Math. 29, 244-269 (1978). [ZBL0382.35043](https://zbmath.org/?q=an:0382.35043).</cite> 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]
<cite authors="Ivrii, Victor">_Ivrii, Victor_, [**100 years of Weyl’s law**](http://dx.doi.org/10.1007/s13373-016-0089-y), Bull. Math. Sci. 6, No. 3, 379-452 (2016). [ZBL1358.35075](https://zbmath.org/?q=an:1358.35075).</cite>