By Fourier transform, it is well known that the Klein-Gordon, wave and Schrödinger flows are $e^{it\sqrt{1-\Delta}}$, $e^{it \sqrt{-\Delta}}$ and $e^{it\Delta}$, respectively.
I saw a phrase "the Klein-Gordon flow behaves like the wave flow at high frequency and the Schrödinger flow at low frequency."
I guess it is true at high frequency because its Fourier transform, $e^{it\sqrt{1+|\xi|^2}}$ and $e^{it|\xi|}$ are similar when $|\xi|$ is large (so we can ignore $1$). But, at low frequency, $|\xi|^2$ and $\sqrt{1+|\xi|^2}$ approach to $0$ and $1$, respectively ($e^{it|\xi|^2}$ and $e^{it\sqrt{1+|\xi|}}$).
Hence I can't understand why it behaves like Schrödinger flow.
Could you explain exactly what it means?