Continuity properties of the Laguerre operator and its propagator

TL;DR

This paper analyzes the continuity properties of the Laguerre operator's propagator, linking it to harmonic oscillators and integral transforms, with a focus on Pilipović spaces for well-posedness.

Contribution

It provides a detailed analysis of the propagator's continuity, connecting Laguerre operators with harmonic oscillators and fractional transforms, and emphasizes Pilipović spaces' role.

Findings

Established continuity properties of the Laguerre propagator

Linked Laguerre operator analysis to harmonic oscillator problems

Connected integral transforms like fractional Fourier and Hankel transforms

Abstract

We study the well-posedness of a Cauchy problem associated with the general form of the Laguerre operator and relate it to the corresponding global problem for the harmonic oscillator. To this end, we carry out a detailed analysis of the continuity properties of the associated propagator. Furthermore, we establish connections between several integral transforms, including the fractional Fourier transform and the fractional Hankel transform. Our results highlight the role of Pilipovi\'c spaces on positive orthants when studying problems involving the Laguerre operator.