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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 annhilation 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 necessart truncation of the Fock basis has to be taken to approch 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 $\mathcal{\hat{G}}$.

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    I think this question should be asked on physics stack exchange. There you can find plenty of people who actually understand your notation and terminology. – 2017-02-22
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    @Blazej, advice taken thank you! – 2017-02-22
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    [Cross-posted to Physics Stack Exchange](http://physics.stackexchange.com/questions/313970/finding-the-eigenvalues-and-eignevectors-of-this-operator). – 2017-02-22

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