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Fractional Fourier transforms, powers of harmonic oscillators and their propagators on Pilipovic and modulation spaces

Joachim Toft, Divyang G. Bhimani, Ramesh Manna

Published Nov 18, 2021• 0 citations
OpenAlex

Abstract

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces. Especially we extend some known results in the literature. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.

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We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces. Especially we extend some known results in the literature. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.
The N x N Fourier matrix is one distinguished element within the group U(N) of all N x N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N-->infinity contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.
The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.
Abstract The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.
Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.