Example of strongly continuous group of operators I have to check if the collection of operators $\{ A(t)\}_{t \in R}$ is a strongly continuous unitary group, where:
$$A(t):L^2(R)\rightarrow L^2(R)$$
$$(A(t)\psi)(x)=e^\frac t 2 \psi(e^tx)$$
I've already checked that:
1)  $\forall t \in R : $ $A(t)$ is a unitary operator;
2) $A(0)=\mathbb{I }$
3) $A(t+s)=A(t)A(s)$
I still have to show that the collection of operator is strongly continuous:
4) $ \forall \psi \in L^2(R): \quad A(t)\psi \rightarrow \psi \quad$ as $t\rightarrow 0$.
How can I prove this?
 A: You may want to consider applying Stone's theorem on one-parameter unitary groups which establishes a one-to-one connection between strongly continuous unitary groups and a generator, i.e. a (possibly unbounded) operator $H$ acting on some dense domain $D(L^2(\mathbb R))\subset L^2(\mathbb R))$ such that $$A(t)=e^{itH}$$ for all $t\in\mathbb R$.
It turns out that your "collection of operators" $\lbrace A(t)\rbrace_{t\in\mathbb R}$ simply is the squeezing operator generated by $$H_A=\frac12 (\hat x\hat p+\hat p\hat x)\,,$$ refer to, e.g., Section IV & VIII in this article (click here for the arXiv version). This $H_A$ involves the position operator
$$
\hat x: D(\hat x) \to L^2(\mathbb R) \qquad \psi(x)\mapsto x\psi(x)
$$
and the momentum operator (set $\hbar=1$)
$$
\hat p: D(\hat p) \to L^2(\mathbb R) \qquad \psi(x)\mapsto -i\psi'(x)
$$
on appropriate domains $$\mathcal S(\mathbb R)\subset D(H_A)\subset D(\hat x)\cap D(\hat p)\subset L^2(\mathbb R)$$ which contain the Schwartz space and are chosen such that $\hat x,\hat p,H_A$ are self-adjoint. Then one readily verifies that
$$
e^{itH_A}\psi(x)=e^{t/2}e^{tx\frac{d}{dx}}\psi(x)=e^{t/2}\psi(e^tx)=(A(t)\psi)(x)
$$
for all $t\in\mathbb R$ which in turn by Stone's theorem implies strong continuity of the unitary group in question.
