I got interest in the field of Quantum Harmonic Analysis and would like to work on this field. For that I would like to know the recent trends in this field in mathematical point of view.
In phase space (position-momentum/time-frequency) there is a classical reference of Werner,
R. F. Werner. Quantum harmonic analysis on phase space. J. Math. Phys., 25(5):1404–1411, 1984.
This has inspired some recent work in time-frequency localization operators of higher rank. Check the work of Luef and Skrettingland and developments around this, in particular (but not restricted to),
F Luef, E Skrettingland, Mixed-state localization operators: Cohen’s class and trace class operators, Journal of Fourier Analysis and Applications 25, 2064-2108, 2019.
F Luef, E Skrettingland, On accumulated Cohen’s class distributions and mixed-state localization operators, Constructive Approximation 52, 31-64, 2020.
Skrettingland, E. (2020). Quantum harmonic analysis on lattices and Gabor multipliers. Journal of Fourier Analysis and Applications, 26, 1-37.
Luef, Franz, and Eirik Skrettingland. "Convolutions for localization operators." Journal de Mathématiques Pures et Appliquées 118 (2018): 288-316.
There is an upcoming workshop soon, maybe it still possible to at least attend (perhaps online), you can try to contact the organizers:
https://www.ntnu.edu/imf/qha2023
Harmonic anaysis is now generalized into wavelet theories. For example step functions in place of sine,, cosine.
Im surprised you ask about the Quantum version. The only features that distingish that is the presence of
h, the Planck constant and the phsical interpretation of the square modulus of the compex valued wave function.
The Fourier transformations are then used to show the ucertainty principle, that sais essentially that a very extended wave function needs only a few wave lengths; or a sharply located one needs many. This result spills over into many other situations, without mention of h.
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