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Its power is amazing. For a Hamiltonian, you define the Green function as

$$G = \frac{1}{\lambda E-H} .$$

Who first come up with this definition? What was the motivation?

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  • $\begingroup$ Probably someone named Green $\endgroup$ – tox123 Mar 27 '16 at 20:19
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This is not quite the definition of the Green function, but rather of the resolvent, more traditional notation is $R_{\lambda}:=(\lambda E-H)^{-1}$ with $H$ the Hamiltonian of the system and $E$ the identity operator. When $H$ is a differential operator, as the Schrodinger operator with some boundary conditions for example, this resolvent can be represented by an integral operator, whose kernel is called the Green function, i.e. $(R_{\lambda}f)(x)=\int G_{\lambda}(x,y)f(y)\,dy$. The resolvent formalism for integral operators was originally developed by Fredholm in 1903, the name is due to Hilbert.

As for the Green function, Green introduced it back in 1828 for solving boundary value problems, and there was no need to "introduce" it into quantum mechanics specifically. The kinds of boundary value problems that come up there are essentially the problems that physicists and mathematicians were solving in the context of classical mechanics throughout 19th century (e.g. the Sturm-Liouville problems), and Schrödinger, Dirac, Weyl, etc., were well familiar with the Green function method of solving them in 1920s, when quantum mechanics was created. The same goes for the Fourier transform methods and the operational calculus.

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