Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates
Välkommen till föreläsningen i seminarieserien i matematik.
We show 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 [1, 2, 3, 4]. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovi ́c spaces. Especially we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are continuous on any Pilipovi ́c space and corresponding distribution space, which are not Gelfand-Shilov spaces.
The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.
References
[1] D. G. Bhimani, The nonlinear Schro ̈dinger equations with harmonic dinger equations with harmonic potential modulation spaces, Discrete Contin. Dyn. Syst. 39 (2019), 5923–5944.
[2] D. Bhimani, R. Balhara, S. Thangavelu Hermite multipliers on modulation spaces, in: Analysis and partial differential equations: perspectives from developing countries, Springer Proc. Math. Stat., 275, Springer, Cham, 2019, pp. 42–64.
[3] E. Cordero, K. H. Gro ̈chenig, F. Nicola, L. Rodino Generalized metaplectic operators and the Schro ̈dinger equation with a potential in the Sjo ̈strand class, J. Math. Phys. 55 (2014), 081506.
[4] E. Cordero, F. Nicola Metaplectic representation on Wiener amalgam spaces and ap- plications to the Schr ̈odinger equation, J. Func. Anal. 254 (2008), 506–534.
[5] J. Toft, Images of function and distribution spaces under the Bargmann transform, J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.
[6] J. Toft, D. Bhimani, R. Manna Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipovi ́c and modulation spaces, (preprint), arXiv:2111.09575.