Unitary Operator on infinitely differentiable function

Question

If $Ψ(x)$ is an infinitely differentiable function, And the operator $\widehat{D}=\exp(ax\frac{d}{dx})$, then show that $\widehat{D}Ψ(x) = Ψ(e^a x)$

Answer 1

Let $U(a, x):= \hat D_a\Psi(x)$, then we see that \begin{align} \frac{\partial U}{\partial a} = x\frac{\partial U}{\partial x}\ \ \text{ and } \ \ \ U(0, x) = \Psi(x). \end{align} Then we see that \begin{align} \frac{d}{da}U(a, x(a)) = 0 \end{align} provided $x' = -x$. Hence it follows $x = x_0e^{-a}$ and $U(a, x(a)) = U(0, x_0)$. This means \begin{align} U(a, x) = U(0, e^a x) = \Psi(e^a x). \end{align}