There are techniques for solving PDE's, such as Fock-Schwinger method in physics, which involve translating the problem from the language of distributions to the language of the abstract Hilbert spaces.
For example, the Green's function equation is: $$L(i\partial_x,x)G(x,x')= \delta(x-x')$$
A typical approach to deal with such equations is to note that we can choose vectors in Hilbert space as: $$\langle x|x'\rangle = \delta(x-x')$$ and, having this in mind, move $x$ and $i\partial_x$ "into" Hilbert space according to the rule: $$i\partial_x \langle x|x'\rangle = \langle x|P|x'\rangle [1]$$ $$x \langle x|x'\rangle = \langle x|Q|x'\rangle [2]$$ where $P$ is chosen to commute with $Q$ as $[P,Q]=i * I$.
Therefore, inside bra-ket a linear operator $L$ is represented as: $$L(i\partial_x,x)\langle x|x'\rangle = \langle x|L(P,Q)|x'\rangle $$ and analytic functions of $L$, such as $exp(itL)$, will move into bra-ket with some additional terms due to commutation.
My question is the following. On what mathematical grounds can we use rules [1] and [2]? I see that they are somehow related to the Stone - von Neumann theorem, but I don't see the exact justification. Moreover, $\langle x|x'\rangle $ is being normalized in terms of distributions, not scalars, that is kind of a generalization to the usual notion of vector spaces with dot product, which also needs justification.
Any textbook on Hilbert spaces or Quantum Mechanics that clarifies these rules? (I mean, in a more or less strict way, not "on the physical grounds") Is there any other book which highly exploits this Hilbert space technique to solve different kind of problems?
I proved similar relations in terms of distributions: $$\delta(x-x') = \int dX \space \delta(X-x) \space \delta(X-x')$$ $$x \space \delta(x-x') = \int dX \space \delta(X-x) \space x \space \delta(X-x')$$ and by definition of the derivative in the space of distributions $$\partial_x \delta(X-x) \to -\partial_X \delta(X-x) \to +\delta(X-x)\partial_X$$ I finally get: $$i\partial_x \space \delta(x-x') = \int dX \space \delta(X-x) \space i\partial_X \space \delta(X-x')$$ which looks like a dot product in Hilbert space if with make a map: $$\delta(X-x) \to |x\rangle $$ $$\int dX \space \delta(X-x) \space \delta(X-x') \space \to \space \langle x|x'\rangle $$
And also, I have another question. Is there a proof that also justifies this kind of a map between distribution-like objects to Hilbert space vectors?