Question

I'm very interested in the properties of the semigroup $e^{it\sqrt{-\Delta}}$, it may has some fundamental differences(such as the kernel) with the well-known schrodinger semigroup $e^{it\Delta}$.

Any properties (or references or books) that related this semigroup are appreciated.

Thanks!

Answer 1

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. In particular, it has a finite propagation speed. This can also be seen from the dispersion relation $\omega = |\xi|$, 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.