Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and fractional Fourier
transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces,
and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on
modulation spaces. Especially we extend some results in our recent works and those of Bhimani, Cordero,
GrĂ¶chenig, Manna, Thangavelu, and others. We also show that general forms of fractional harmonic oscillator
propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially
we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are
continuous on any Pilipovic space and corresponding distribution space, which are *not* Gelfandâ€“Shilov spaces.
(The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.)

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Last updated: 28 Nov 2023