Continuity of Green's function

[Wledig]

Why can't G and its derivative be continuous in the relation below?

$$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$

 

[Orodruin]

They can, but not if it is to hold as $\epsilon \to 0$.

 

[Wledig]

Yes, I realized that. Both terms would vanish if that was the case, however the way this situation is avoided by letting dG/dx have a discontinuity at x = t seems rather arbitrary. Couldn't the same be accomplished by letting G have a discontinuity in t? I guess I just find a bit awkward to let G be continuous but not its derivative, not sure if this is a condition that should hold true.

 

[Orodruin]

Couldn't the same be accomplished by letting G have a discontinuity in t?

No, that would not help. The integral would still go to zero.

I guess I just find a bit awkward to let G be continuous but not its derivative, not sure if this is a condition that should hold true.

This is really not a strange thing. There are many examples of sich functions. For example $f(x) = |x|$.

Still, I can relate to your worries and I also never liked this argument although it does its job in the end. My preferred argument, and the one I use in my book, is to write $G$ as a piecewise function $G(x,x’) = \theta(x-x’)g_1(x) + \theta(x’-x)g_2(x)$ and insert this into the differential equation using that $\theta’$ is a delta function. You will need some delta function properties, but once the smoke clears you will obtain exactly the same result.

 

[Wledig]

This is really not a strange thing. There are many examples of sich functions. For example f(x)=|x|f(x)=|x|f(x) = |x|.

You're right, I completely forgot this classic example.

Still, I can relate to your worries and I also never liked this argument although it does its job in the end. My preferred argument, and the one I use in my book, is to write GGG as a piecewise function G(x,x′)=θ(x−x′)g1(x)+θ(x′−x)g2(x)G(x,x′)=θ(x−x′)g1(x)+θ(x′−x)g2(x)G(x,x’) = \theta(x-x’)g_1(x) + \theta(x’-x)g_2(x) and insert this into the differential equation using that θ′θ′\theta’ is a delta function. You will need some delta function properties, but once the smoke clears you will obtain exactly the same result.

That's actually a much more natural way of thinking about the problem, since the derivative of the step function is Dirac's delta, we're just left with the boundary conditions. I just wonder how to reach the relation below from it though. $$ \lim_{\epsilon \rightarrow 0} \left( \dfrac{dG}{dx} \Big|_{x=t+\epsilon} - \dfrac{dG}{dx} \Big|_{x=t-\epsilon} \right) = \dfrac{1}{p(t)} $$

 

[Orodruin]

What do you get when you do the insertion into your differential equation?

 

[Wledig]

Sorry, I'm confused. I tried inserting it into $\dfrac{d}{dx}\left(p(x)\dfrac{dG}{dx}\right)+q(x)G=\delta(x-t)$ but I got a pretty long expression in return that didn't seem like it could be simplified into anything that would help me reach the relation. Am I following your line of thought correctly? Maybe I misunderstood something.

 

[Orodruin]

Am I following your line of thought correctly?

It is impossible to know that unless you actually show us what you got.

 

[Wledig]

Alright, here's what I got:
$$ p[(g''_1 \theta(x-x') + 2g'_1 \delta(x-x') + g_1 \delta(x-x')') + (g''_2 \theta(x'-x) + 2g'_2 \delta(x'-x) + g_2 \delta(x'-x)')] + p'(g'_1 \theta(x-x') + g'_2\theta(x'-x) + g_1 \delta(x-x') + g_2 \delta(x'-x)) + q(g_1 \theta(x-x') + g_2\theta(x'-x)) = \delta(x-x')$$
Now, I looked it up that the derivative of the delta function is $\frac{\delta}{x}$ so I could substitute it there, but it doesn't seem to lead anywhere. I also thought about isolating $\frac{dG}{dx}$ and inserting in the other terms to come up with something, but that didn't seem to work either.

 

[Orodruin]

The derivative of the $\delta$ is a separate distribution in and of itself. It does not really make sense to say that it is $\delta/x$. Its exact properties are not important at the moment, but know that it is separate from the delta itself. Before you take the second derivative, note that $f(x) \delta(x-x') = f(x') \delta(x-x')$. That will save you some annoying cross terms that are a bit tricky to handle. After that, start collecting terms with the same factors of $\theta$, $\delta$, or $\delta'$ and identify it with the corresponding terms on the right-hand side.