The operator

$$\mathcal{\hat{G}} = (\xi - 1) \sum_{j=1}^N \int dk_j \; k_j \hat{a}^\dagger_j(k_j)\hat{a}_j(k_j),$$

is physically similar to the momentum operator in quantum mechanics. It has the same units and generates spatial shifts. It differs only by generating spatial shifts in a system according the some deforming function $\xi$. Note that $N$ is a positive integer than can be arbitrarily large (it describes the number of particles which consititute the system).

I would like to find the eigenvalues and eigenvectors of the operator. How would I do this? I know that the creation and annihilation operators in the Fock basis can be written in their matrix representation as shown here. I am unsure on how to proceed from this point. Any help or advice would be appreciated. If necessary truncation of the Fock basis has to be taken to approach the problem, then this would be fine.

The aim:

The reason for wanting to decompose the operator in its matrix form and determine its eigenvalues and eigenvectors is since I would like to find the difference between the maximum and minimum eigenvalues of $\mathcal{\hat{G}}$.

Insight into problem

Consider a 1D array of equidistant sources, each independent, such that $\vert\Psi\rangle = \otimes_{j=1}^N\vert\psi\rangle_j$. The positions of each of the sources are encoded in the state. Deforming the array by means of a homogeneous deformation $\xi$, the new state can be determined by the operation of the unitary $\hat{U}$ on the old state, where $U$ is a function of $\xi$.

Cross-posted from Math Stack Exchange

  • $\begingroup$ This question was originally posted in Math Stack Exchange. With advice, I have posted it here too. $\endgroup$ – Sid Feb 22 '17 at 12:54
  • $\begingroup$ What is $\xi$, why do you describe it as a function, and what does it depend on? $\endgroup$ – Emilio Pisanty Feb 22 '17 at 13:07
  • $\begingroup$ @EmilioPisanty, context provided. $\endgroup$ – Sid Feb 22 '17 at 13:43
  • $\begingroup$ so... $\xi$ could just be omitted, then? If all you have is $\hat A=(\xi-1)\hat B$, and you want to study the effect of $\xi$ on $\hat A$, there isn't really much to say. If the situation is nontrivial for other reasons, you should specify them. $\endgroup$ – Emilio Pisanty Feb 22 '17 at 13:46
  • $\begingroup$ @EmilioPisanty, The factor $\xi$ should really just be seen as some context in the generator given but is unimportant for the question. My question is on how to find the eigenvectors and eigenvalues of the generator G given. $\endgroup$ – Sid Feb 22 '17 at 14:31

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