The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta)(\partial_{t}+i\sqrt{-\Delta})$$ From the above that a wave operator can be seen as the product of two schr$\ddot{o}$dinger operators.Indeed,compared with other kinds of differential operators(elliptic,parabolic),these two operators share more in common.Such as they don't have the usual global regularity property,and the use of harmonic analysis has made a great success in these two areas.

So I'm very curious to know the intersting links or fundemental differences between this two operator in order to understand them better.