The question is about Theorem 14.1 in Teschl's Partial Differential Equations which goes like that
$\textbf{Theorem 14.1.}$ The family $T_S(t)$ is a $C_0$-group in $H^{r}(\mathbb{R}^{n})$ whose generator is $i∆$, $D(i∆) = H^{r+2}(\mathbb{R}^{n})$.
Earlier on, the family $T_S(t)$ is defined as $\mathcal{F}^{-1}e^{-i\lvert p \rvert^{2}t}\mathcal{F}$ which is just a direct consequence of computing the solutions for the linear Schrödinger equation by using the Fourier transform, i.e. our solutions look like $u(x,t) = T_S(t)u_0(x)$.
Now, the motivation to get to our generator $A = i∆$ is that we can depict the solutions of the Schrödinger equation as $u(x,t) = u_0(x)e^{it∆}$, like in the link below:
Schrödinger operator: where is the generator to be defined?
However, I do not really understand how we do come up with this solution. Especially the term $e^{it∆}$ doesn't really make sense to me. Could anybody explain, how to read that expression and how we do come up with this generator for our problem?
Furthermore, I do not understand why $D(A) = H^{r+2}(\mathbb{R}^n)$ should hold (so the domain of the generator is the Sobolev space with coefficient $r+2$). Can anybody explain me that, too?
Time evolution is an exponential operation in the time variable. It's an odd fact of nature that needs some explanation. In order to explain, suppose you have a system whose state is described as a vector $x$ in some Hilbert space $\mathscr{H}$ at a particular time. An evolution operator $E(t_2,t_1)$ acts on a known state vector $x$ at time $t_1$ and "evolves" it to the new state vector $y=E(t_2,t_1)x$ at time $t_2$. By definition $E(t_3,t_2)E(t_2,t_1)=E(t_3,t_1)$, which is an exponential type of property. For time invariant systems, the evolution depends only on the differences in time. More explicitly, $E(t_2,t_1)=\scr{E}(t_2-t_1)$. $$ \scr{E}(t_3-t_2)\scr{E}(t_2-t_1)=\scr{E}(t_3-t_1). $$ So, a time evolution operator on a time-invariant system has a true exponential property: $$ \mathscr{E}(0)=I \\ \mathscr{E}(t)\mathscr{E}(t')=\mathscr{E}(t+t'),\;\; t,t' \ge0. $$ Stability is assumed, meaning that the following holds for all state vectors $x$: $\lim_{h\downarrow 0}E(t+h,t)x=x$. It is this assumption that allows you to talk about a generator.