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, 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 $$\left\{ \begin{array}{ll} u_{tt}+\Delta u=0 , & \\ u|_{t=0}=\delta(x-y),\qquad u|_{t=0}=0 & \end{array} \right.$$

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 $e(x,y,\lambda)$ ? I find many books but get nothing, can someone give some reference about the theory of spectral function for elliptic operators? Thank you very much!