We start from an abstract state vector $
\newcommand{\ket}[1]{|{#1}\rangle} \ket{\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form
$$
\DeclareMathOperator{\dif}{d \!}
\newcommand{\ramuno}{\mathrm{i}}
\newcommand{\exponent}{\mathrm{e}}
\newcommand{\bra}[1]{\langle{#1}|}
\newcommand{\braket}[2]{\langle{#1}|{#2}\rangle}
\newcommand{\bracket}[3]{\langle{#1}|{#2}|{#3}\rangle}
\newcommand{\linop}[1]{\hat{#1}}
\newcommand{\dpd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dod}[2]{\frac{\dif{#1}}{\dif{#2}}}
\ramuno \hbar \dod{}{t}
\ket{\Psi(t)} = \hat{H} \ket{\Psi(t)} \, . \quad (1)
$$
Now if we move to position representation of the state vector what will happen to the Schrödinger equation?
In Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues And Linear Vector Spaces by John David Jackson I found the following information (pp. 77-78).
By taking an inner product of both sides of (1) with $\ket{x}$ and using the resolution of identity $\linop{I} = \int\nolimits_{-\infty}^{+\infty} \ket{x'} \bra{x'} \dif{x'}$ on the right side we obtain
$$
\ramuno \hbar \dod{}{t} \braket{x}{\Psi(t)} = \int\limits_{-\infty}^{+\infty} \bracket{ x }{ \linop{H} }{ x' } \braket{ x' }{ \Psi(t) } \dif{x'} \, .
$$
We then introduce a wave function $\Psi(x,t) \equiv \braket{x}{\Psi(t)}$ and if I understand correctly $\bracket{ x }{ \linop{H} }{ x' }$ is also replaced by a function $h(x, x')$ which will lead us to
$$
\ramuno \hbar \dod{}{t} \Psi(x, t) = \int\limits_{-\infty}^{+\infty} h(x, x') \Psi(x', t) \dif{x'} \, .
$$
Now we are one step away from the familiar Schrödinger equation in position representation: we need Hamiltonian operator in position representation $\linop{H}(x, \frac{\hbar}{\ramuno} \dod{}{x})$ to be given by
$$
\linop{H}(x, \frac{\hbar}{\ramuno} \dod{}{x}) \Psi(x, t) = \int\limits_{-\infty}^{+\infty} h(x, x') \Psi(x', t) \dif{x'} \, .
$$
Author claims (p. 44) that
For our purposes the general linear operator $K$ can be written in the explicit form
$$
g = K f \rightarrow g(x) = \int\limits_{a}^{b} k(x,x') f(x') \dif{x'} \quad(2)
$$
The function $k(x,x')$ is called the kernel of the operator $K$.
It is not that I do not trust the author, but since my knowledge of mathematics is not great and I've never seen something like (2) before, I'm confused with this "for our purposes".
What does it actually mean? Is (2) true for any linear operator or for a particular kind of linear operators, say, for self-adjoint linear operators on Hilbert spaces?
Best Answer
Let me first say that I think Tobias Kienzler has done a great job of discussing the intuition behind your question in going from finite to infinite dimensions.
I'll, instead, attempt to address the mathematical content of Jackson's statements. My basic claim will be that
To see this clearly without having to worry about possible mathematical subtleties, let's first consider
Finite dimension
In this case, we can be certain that there exists an orthnormal basis $\{|n\rangle\}_{n=1, \dots N}$ for the Hilbert space $\mathcal H$. Now for any state $|\psi(t)\rangle$ we define the so-called matrix elements of the state and Hamiltonian as follows: \begin{align} \psi_n(t) = \langle n|\psi(t)\rangle, \qquad H_{mn} = \langle m|H|n\rangle \end{align} Now take the inner product of both sides of the Schrodinger equation with $\langle m|$, and use linearity of the inner product and derivative to write \begin{align} \langle m|\frac{d}{dt}|\psi(t)\rangle=\frac{d}{dt}\langle m|\psi(t)\rangle=\frac{d\psi_m}{dt}(t) \end{align} The fact that our basis is orthonormal tells us that we have the resolution of the indentity \begin{align} I = \sum_{m=1}^N|m\rangle\langle m| \end{align} So that after taking the inner product with $\langle m|$, the write hand side of Schrodinger's equation can be written as follows: \begin{align} \langle m|H|\psi(t)\rangle = \sum_{m=1}^N\langle n|H|m\rangle\langle m|\psi(t)\rangle = \sum_{m=1}^N H_{nm}\psi_m(t) \end{align} Equating putting this all together gives the Schrodinger equation in the $\{|n\rangle\}$ basis; \begin{align} \frac{d\psi_n}{dt}(t) = \sum_{m=1}^NH_{nm}\psi_m(t) \end{align}
Infinite dimension
With an infinite number of dimensions, we can choose to write the Schrodinger equation either in a discrete (countable) basis for the Hilbert space $\mathcal H$, which always exists by the way since quantum mechanical Hilbert spaces all possess a countable, orthonormal basis, or we can choose a continuous "basis" like the position "basis" in which to write the equation. I put basis in quotes here because the position space wavefunctions are not actually elements of the Hilbert space since they are not square-integrable functions.
In the case of a countable orthonormal basis, the computation performed above for writing the Schodinger equation in a basis follows through in precisely the same way with the replacement of $N$ with $\infty$ everywhere.
In the case of the "basis" $\{|x\rangle\rangle_{x\in\mathbb R}$, the computation above carries through almost in the exact same way (as your question essentially shows), except the definitions we made in the beginning change slightly. In particular, we define functions $\psi:\mathbb R^2\to\mathbb C$ and $h:\mathbb R^2\to\mathbb C$ by \begin{align} \psi(x,t) = \langle x|\psi(t)\rangle, \qquad h(x,x') = \langle x|H|x'\rangle \end{align} Then the position space representation of the Schrodinger equation follows by taking the inner product of both sides of the equation with $\langle x|$ and using the resolution of the identity \begin{align} I = \int_{-\infty}^\infty dx'\, |x'\rangle\langle x'| \end{align} The only real mathematical subtleties you have to worry about in this case are exactly what sorts of objects the symbols $|x\rangle$ represent (since they are not in the Hilbert space) and in what sense one can write a resolution of the identity for such objects. But once you have taken care of these issues, the conversion of the Schrodinger equation into its expression in a particular "representation" is just a matter of making the appropriate definitions.