Scaled Dirac Delta function: $ \delta (xe^r - y) $ I was reading on squeezed Gaussian states and stumbled upon this paper:
Equivalence Classes of Minimum-Uncertainty Packets. II.
It is mentioned after Eq. $\left(2\right)$ that
$$
\left\langle x\left\vert\,{{\rm e}^{{\rm i}rxp}}\,\right\vert y\right \rangle =
\delta\left(x{\rm e}^{r} - y\right)
$$
where $\left\vert x \right\rangle , \left\vert y \right\rangle$ are position eigenstates.
How is this relation derived?
 A: OP's equation
$$\begin{align} \langle x | e^{ir\hat{x}\hat{p}/\hbar}| y\rangle 
~=~~& \int_{\mathbb{R}}  \mathrm{d}p ~\langle x | e^{ir\hat{x}\hat{p}/\hbar}| p\rangle ~\langle p | y\rangle \cr
~\stackrel{(B)+(D)}{=}&~ \int_{\mathbb{R}}  \mathrm{d}p ~\frac{e^{ixe^rp/\hbar}}{\sqrt{2\pi\hbar}}~\frac{e^{-iyp/\hbar}}{\sqrt{2\pi\hbar}}\cr
~=~~& \delta(xe^r\!-\!y)
\end{align}\tag{A}$$
follows from
$$ \langle x | e^{ir\hat{x}\hat{p}/\hbar}| p\rangle~=~\frac{e^{ixe^rp/\hbar}}{\sqrt{2\pi\hbar}}.\tag{B}$$
Here we have assumed the CCR
$$ [\hat{x},\hat{p}]~=~i\hbar \hat{\bf 1},\tag{C}$$
and the standard Fourier-transform overlap
$$ \langle x | p\rangle ~=~ \frac{e^{ixp/\hbar}}{\sqrt{2\pi\hbar}},\tag{D}$$
cf. e.g. this Phys.SE post.
Sketched proof of eq. (B): Firstly, note that (B) is true for $r=0$. Secondly, differentiate both sides of eq. (B) wrt. $r$ and see that they satisfy the same first-order ODE. Use that
$$
e^{-ir\hat{x}\hat{p}/\hbar}\hat{p} e^{ir\hat{x}\hat{p}/\hbar}~=~
e^{-ir[\hat{x}\hat{p}/\hbar, \cdot]}\hat{p}~\stackrel{(C)}{=}~e^r\hat{p}.\tag{E}$$
$\Box$
