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 (general type) schr$\ddot{o}$dinger operators.Indeed,compared with other kinds of differential operators(elliptic,parabolic),these two operators seem to 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.

**Edit:**In the above,I'm refering to a general dispersive equation:$u_{t}-i\phi({D})u=0$.When $\phi=|\xi|^{2}$,it's the free schr$\ddot{o}$dinger equation.

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 (general type) schr$\ddot{o}$dinger operators.Indeed,compared with other kinds of differential operators(elliptic,parabolic),these two operators seem to 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.

**Edit:**In the above,I'm refering to a general dispersive equation:$u_{t}-i\phi({D})u=0$.When $\phi=-|\xi|^{2}$,it's the free schr$\ddot{o}$dinger equation.

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 seem to 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.

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 (general type) schr$\ddot{o}$dinger operators.Indeed,compared with other kinds of differential operators(elliptic,parabolic),these two operators seem to 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.

**Edit:**In the above,I'm refering to a general dispersive equation:$u_{t}-i\phi({D})u=0$.When $\phi=|\xi|^{2}$,it's the free schr$\ddot{o}$dinger equation.