A Has Hilbert transform ever been used in Quantum Theory?

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The Hilbert transform has not been widely recognized or utilized in quantum theory, despite its presence in complex analysis and signal analysis courses. Some discussions suggest its application in quantum control theory and scattering theory, particularly through the Jost functions and dispersion relations, where they exhibit Hilbert transform characteristics. The Kramers-Kronig relations are mentioned as potentially relevant, being applicable in various fields, including quantum optics. However, concrete examples of direct usage in quantum theory remain scarce. Overall, the Hilbert transform's role in quantum theory is not well-documented or understood.
mad mathematician
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Anyone knows if this transform ever been used in QT directly?

I just had seen it in one advanced course in complex analysis which I failed and in singals analysis courses in EE.
But in all the books and courses in QT never I had seen this transform being used.

Perhaps in Quantum Control theory...
https://en.wikipedia.org/wiki/Hilbert_transform
 
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In scattering theory, when you consider the so-called (complex-valued) Jost functions ##F_l(k)##, the dispersion relations relate the real and the imaginary parts of ##F_l(k) -1##. And the specific form of these relations make each the Hilbert transforms of the other. Just google for Jost functions and dispersion relations.
 
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I used it once in a perturbation expansion of a unitary operator. It was somewhat simpler than the usual expansion of the complex exponential. But I forgot the details.
 
mad mathematician said:
Anyone knows if this transform ever been used in QT directly?

I just had seen it in one advanced course in complex analysis which I failed and in singals analysis courses in EE.
But in all the books and courses in QT never I had seen this transform being used.

Perhaps in Quantum Control theory...
https://en.wikipedia.org/wiki/Hilbert_transform
The Kramers Kronig relations in (quantum) optics.

Edit: Looks like the KK relations are used for almost everything:
https://en.wikipedia.org/wiki/Kramers–Kronig_relations
 
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Thread 'Angular Momentum Vector and Its Magnitude'

For the quantum state ##|l,m\rangle= |2,0\rangle## the z-component of angular momentum is zero and ##|L^2|=6 \hbar^2##. According to uncertainty it is impossible to determine the values of ##L_x, L_y, L_z## simultaneously. However, we know that ##L_x## and ## L_y##, like ##L_z##, get the values ##(-2,-1,0,1,2) \hbar##. In other words, for the state ##|2,0\rangle## we have ##\vec{L}=(L_x, L_y,0)## with ##L_x## and ## L_y## one of the values ##(-2,-1,0,1,2) \hbar##. But none of these...
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