The physicist [Yoichiro Nambu](https://en.wikipedia.org/wiki/Yoichiro_Nambu) introduced in a 1950 paper [A Note on the Eigenvalue Problem in Crystal Statistics](http://www.totoha.net/archiv/3675.pdf) the notion of an "*eigenoperator*" (page 12, see [Nambu and the Ising model](https://arxiv.org/abs/2209.01122) for a recent discussion of this early work by the 2008 Nobel laureate).


Given a self-adjoint operator $H$, the eigenoperator $X$ satisfies
$$HX-XH=\lambda X,\;\;\lambda\in\mathbb{R}.$$


**Q:** A Google search for "eigenoperator" does not return much, has this notion found its place in the mathematical literature, perhaps under a different name?




---


I append a screenshot of the relevant paragraph from Nambu's paper:


![](https://i.stack.imgur.com/PBtL6.png)
Footnote 6) refers to Nambu's paper [On the Method of the Third Quantization](https://www.semanticscholar.org/paper/On-the-Method-of-the-Third-Quantization.-Nambu/6f3f8b38cf6c6963627cfc883afc580d347b4e95), where $X$ is called an "eigenmatrix".

Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?
Such an $X$ is an *eigenvector of $\,\operatorname{ad}(H)$*. Joint eigenspace decompositions of several $\operatorname{ad}(H\_i)$ are commonplace in math since the work of Lie, Killing, Cartan, with the joint eigenvectors called *root vectors*. So I would say that the notion “had a place” already before Nambu.


