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I have the following density operator $$\frac{1}{t \pi N} \int_{\mathbb{C}} \mathrm{d}^2\gamma \exp \left[ -\frac{|\gamma+r\alpha|^2}{t^2 N} \right] |{\gamma}\rangle\langle{\gamma}|,$$ where $0\leq t,r\leq 1$ are fixed, and $\alpha \in \mathbb{C}$ is also a constant. The integral runs over coherent states, and $\mathrm{d}^2\gamma := \mathrm{d}\Re(\gamma)\mathrm{d}\Im(\gamma)$ (where both real and imaginary parts run from $-\infty$ to $+\infty$ to span the whole of $\mathbb{C}$).

I would like to find the eigenstates of this operator. Although it is already diagonal, I am having trouble identifying the eigensolutions. I hope this is a very silly question and get a quick reply!

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    $\begingroup$ Isn't this diagonal in the gamma basis already? $\endgroup$ – Pratik Rath Jun 15 at 13:04
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    $\begingroup$ @PratikRath. Coherent states are not orthogonal, so they do not compose a basis (although they are complete). $\endgroup$ – march Jun 20 at 16:05

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