Consider the Eqn. $$D|f\rangle=0.\tag{1}$$ I want to express it as a differential equation $$\mathcal{D}_x f(x)=0\tag{2}$$ where $\mathcal{D}_x$ is the differential operator representation of the operator $D$. We get from (1), $$\langle x|D|f\rangle=0.$$ Introducing a complete set of position eigenbasis $\{|x\rangle\}$ (with $\langle x|x^\prime\rangle=\delta(x-x^\prime)$ and $\int dx|x\rangle\langle x|=\bf{1}$), we get, $$\int dx^\prime \langle x|D|x^\prime\rangle\langle x^\prime|f\rangle=0.\tag{3}$$ $$=\int dx^\prime \langle x|D|x^\prime\rangle f(x^\prime).$$

  1. How should I proceed next to integrate the $x^\prime$ integral and derive Eqn. (2). Using the equation $DD^{-1}=\bf{1}$ doesn't seem to help.

  2. If someone can explain how to write the Eqn.(2) in the abstract notation of Eqn. (1) that will be equally helpful.

  • $\begingroup$ What's the form of $\langle x|D|x^\prime\rangle$, maybe in position eigenbase, the operator has the form you need. $\endgroup$ – Rafa Budría Mar 26 '17 at 8:51
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    $\begingroup$ Do you have the commutation relation between $D$ and $x$. If you have one, then it is possible to write the representation of $D$ in $x$ space. It is not possible, however, to represent an algebra if you do not have anything which defines it) $\endgroup$ – Kiryl Pesotski Mar 28 '17 at 13:35

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