Talk:Green's function

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1 Page name
2 All functions are equal, and discontinous functions aren't less equal than others, are they?
3 Applications to Quantum Mechanics
4 bad form
5 Minus sign missing
6 More Examples
7 Green's functions in condensed matter physics
8 Intuition
9 kernel
10 Title

[edit] Page name

People, let's call it a Green function. This use of Green's is killing me!

I've always heard it called a Green's function. This case seems somewhat specific, would it be possible to generalise it to a degree n? - 3mta3 01:23, 20 Jun 2005 (UTC)

It should be 'Green function': compare Hermite polynomial, Abelian Group, Grothendieck topology, etc.--88.104.215.243 11:05, 9 July 2006 (UTC)

I'm going to rename the page accordingly. AMS 108, Applied Functional Analysis by Zeidler calls it "Green function". —Ben FrantzDale 00:17, 13 November 2006 (UTC)

You've renamed the page, but left the references to Green's function in the article. Not very thorough. In my experience, most of the standard texts (e.g., Morse and Feshbach) refer to it as Green's function. When I studied the Green's function, the professor specifically mentioned the unusual use of the possessive. His explanation was that to call it the 'Green function' sounded as if you were talking about a function that was colored green in some way, which tended to confuse people, especially students.

--Tex 16:22, 8 February 2007 (UTC)

[edit] All functions are equal, and discontinous functions aren't less equal than others, are they?

The article says "Green's functions are distributions in general, not functions, meaning they can have discontinuities.". This cannot possibly be correct, can it? I mean, a function having discontinuities sure doesn't make it any less of a function. The author must have had something else in mind. Please rephrase. Jonas Olson 21:36, 27 September 2005 (UTC)

Umh. Not totally satisfactory formulation. More discontinous than any function, so to say, like Dirac delta function, which isn't a function. --Pjacobi 23:01, 27 September 2005 (UTC)

You raise a good point about the Green's being called stricly a distribution, and not a 'true' function. What of the case of a dimensionless operator L=1? Would not the Green's be identified with the Dirac Delta (assuming infinitesimal integration)? I.E. - it is only a function in the sense that it acts locally on said operator.

[edit] Applications to Quantum Mechanics

I'd love to see an elaboration of the applications to QM. --User:TobinFricke

A couple of references I would suggest mentioning here are the books Green Functions for Ordered and Disordered Systems (Studies in Mathematical Physics) by Antonios Gonis (# ISBN-10: 0444889868 or # ISBN-13: 978-0444889867) and Green's Functions in Quantum Physics by Eleftherios N. Economou (# ISBN-10: 3540288384 or # ISBN-13: 978-3540288381)

The relationship between the Schrödinger Green's function and the local density of states is essentially the Plemelj-Sokhotsky equations (for the resolvent).

Anyway, it would be nice to have a better expert than I to fill in these details.

[edit] bad form

"we leave it to the reader to fill in the in-between steps" Is something contained in bad textbooks and bad lectures. Can anyone fill in the blanks here?

[edit] Minus sign missing

shouldn't there be a minus sign in the Laplacian Green's function 1/|x-x'| (and in the expressionwith the charge density \rho)? maybe I'm confused at the moment, but I'm pretty sure there should be... If no one answers I'll change it. Dan Gluck 16:35, 10 October 2006 (UTC)

[edit] More Examples

I think we all know the best way to learn stuff is Examples, Like actual problems. I think it would be great if we could get many examples from some text books or somewhere, to give readers (and hopless students) an idea of how to use the this function. this is a more General statement to the applications to Quantum Mechanics Section.24.25.211.241 04:48, 24 October 2006 (UTC)

[edit] Green's functions in condensed matter physics

I think there's a little confusion with regards to this section. Until someone disagrees (Andywall?), I'd like to replace it with a section explaining the connections and differences between this kind of Green('s) function — the inverse of a linear operator — and what's meant by the term in quantum/statistical field theory, where it essentially just means correlation function. Stevvers 03:15, 5 November 2006 (UTC)

I have added a sentence in the introduction to mention this distinction and will add something to Correlation function (quantum field theory).Stevvers 01:16, 10 November 2006 (UTC)

[edit] Intuition

After looking at my math text, I think I just "got" this topic, at least for the case of $- u'' = f$ . I would add my understanding to the page, but am not sure it is a complete understanding. Is this right?:

The Green function, $\mathcal{G}(x,y)$ , can be thought of as the response of the solution variable, u, to a delta function,

δ(x - y)

. Because the system is linear, we can apply superposition: we can represent our given f as the sum of delta functions and expect our solution to be a sum of the responses to each of those delta functions.

For example, suppose we have a horizontal string with distributed and point weights on it and we want to know its deflection along its length. Assume small deflection (so that the problem is linear). We could hang a unit mass at every point, y, along the string and record the deflection as a function of x. This would be our $\mathcal{G}(x,y)$ . (Obviously we would expect a V shape centered on the point). The the solution will be

$u(x) = \int_a^b \mathcal{G}(x,y) f(y)\,dy$ .

Is that about right? —Ben FrantzDale 00:16, 13 November 2006 (UTC)

Yes, that sounds exactly right. If you put this into the page, make sure you explicitly write the differential equation that the string satisfies. Stevvers 21:27, 15 November 2006 (UTC)

[edit] kernel

Currently "kernel" links to the "kernel (algebra)" article i.e. the article on the kernel of a homomorphism. I don't see the connection between the two uses of the word "kernel," and I'm wondering if the link should be removed.

Atmd 01:33, 29 January 2007 (UTC)

The usage of the term kernel in the article is correct. What second use of the word, besides the one at kernel (algebra), are you referring to ? MathMartin 18:46, 7 February 2007 (UTC)

A kernel function? I don't believe a green's function is an abelian group. I'm confused as well. Sim 05:17, 1 March 2007 (UTC)

I am confused by you confusion :) So let me try to clear this up. A green function is associated to a linear operator L. Linear operators are group morphism (among other things) and thus we can define a kernel for them which is exactly what is done at Kernel_(algebra)#Linear_operators. Perhaps you could describe more clearly what exactly in the text confuses you so we can try to clear things up ? MathMartin 15:04, 21 March 2007 (UTC)

[edit] Title

The article calls it "Green's Function" throughout (aside from a rather obscurely-written lead, which I've now changed), and all the sources given call it "Green's Function". There seems to be no good reason for us not to call it by that name. (Googling supports the "Green's" versus "Green" name.) --Mel Etitis (Talk) 09:23, 1 March 2007 (UTC)

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