A Has Hilbert transform ever been used in Quantum Theory?

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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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
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