I Continuity of Green's function

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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

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

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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.
 
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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

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

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Am I following your line of thought correctly?
It is impossible to know that unless you actually show us what you got.
 
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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

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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.
 

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