- #1

hiyok

- 109

- 0

Hi,

Let me describe the question. Suppose you have two distinct solutions, say, [itex]G_{1}(z)[/itex] and [itex]G_{2}(z)[/itex], to such a linear differential equation, [itex](\partial^2_z-q^2)G(z)=\delta(z)[/itex], where [itex]\delta(z)[/itex] denotes the Dirac function and [itex]q^2[/itex] is a constant. Now I'd like to evaluate this integral: [itex]\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)[/itex]. I will use this property: [itex]f(z)\delta^{n}(z)=(-1)^{n}f^{n}(z)\delta(z)[/itex]. Thus, if I in the first place integrate over [itex]z_1[/itex], I should then find

[tex]\begin{eqnarray*}

\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1G_1(z-z_1)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)\\

& = & \int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1(\partial^2_{z_1}-q^2)G_1(z-z_1)\delta(z_1-z_2)\\

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1\delta(z-z_1)\delta(z_1-z_2)\\

& = & G_2(z-z')

\end{eqnarray*}[/tex] On other hand, if it is first integrated over [itex]z_2[/itex], one would instead find [itex]G_1(z-z')[/itex], which differs from previous result. I don't know how to make sense of such ambiguity. Can anybody come to help ?!

Thank you in advance.

hiyok

Let me describe the question. Suppose you have two distinct solutions, say, [itex]G_{1}(z)[/itex] and [itex]G_{2}(z)[/itex], to such a linear differential equation, [itex](\partial^2_z-q^2)G(z)=\delta(z)[/itex], where [itex]\delta(z)[/itex] denotes the Dirac function and [itex]q^2[/itex] is a constant. Now I'd like to evaluate this integral: [itex]\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)[/itex]. I will use this property: [itex]f(z)\delta^{n}(z)=(-1)^{n}f^{n}(z)\delta(z)[/itex]. Thus, if I in the first place integrate over [itex]z_1[/itex], I should then find

[tex]\begin{eqnarray*}

\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1G_1(z-z_1)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)\\

& = & \int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1(\partial^2_{z_1}-q^2)G_1(z-z_1)\delta(z_1-z_2)\\

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1\delta(z-z_1)\delta(z_1-z_2)\\

& = & G_2(z-z')

\end{eqnarray*}[/tex] On other hand, if it is first integrated over [itex]z_2[/itex], one would instead find [itex]G_1(z-z')[/itex], which differs from previous result. I don't know how to make sense of such ambiguity. Can anybody come to help ?!

Thank you in advance.

hiyok

Last edited by a moderator: