Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipovic and modulation spaces

Abstract

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates 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 by Balhara, Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.

38 pages. From version 4 to 5: Improvement of fractional powers of harmonic oscillator and their propagators on Hilbert spaces. From version 3 to 4: Several changes have been performed. Especially we have generalized some results to include multiple ordered fractional Fourier transforms and harmonic oscillator propagators. More references to literature, especially quantum theory and optics