What do we know about the semigroup $e^{it\sqrt{-\Delta}}$ I am very interested in the properties of the semigroup $e^{it\sqrt{-\Delta}}$, which may has some fundamental differences (such as the kernel) with the well-known Schrödinger semigroup  $e^{it\Delta}$.
Any properties (or references or books) that related this semigroup are appreciated.
Thanks!
 A: The wave operator decomposes as
$$
\partial_t^2-\Delta = (\partial_t-i\sqrt{-\Delta})(\partial_t+i\sqrt{-\Delta}),
$$
so you can think of $e^{it\sqrt{-\Delta}}$ as solving a "half of" the wave equation. The dispersion relation is $\omega = |\xi|$, where $\omega$ and $\xi$ are the Fourier variables for $t$ and $x$, respectively. On the other hand, the Schrödinger propagator $e^{it\Delta}$ has the dispersions relation $\omega=|\xi|^2$, which makes it genuinely dispersive, i.e., the propagation speed depends on the frequency.
Note that $e^{it\Delta}$ is not the heat semigroup, which the other answers and comments seem to suggest.
A: You can say a lot about the function $u(t,x)=e^{it\sqrt{-\Delta}}f$ by noticing, as remarked by others, that $u$ is also solution of the Cauchy problem
$$
u_{tt}-\Delta u=0,\quad
u(0,x)=f,\quad
u_t(0,x)=i\sqrt{-\Delta}f.
$$
Thus most estimates for solutions of the wave equation carry over to $u$. However there are some subtle differences, mostly due to the non--locality of the operator $\sqrt{-\Delta}$.
