Abstract
AbstractIn this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$where$\tau :{\open R}^n\to {\open R}^n$is a general function. In particular, for the linear choices$\tau (x)=0$,$\tau (x)=x$and$\tau (x)={x}/{2}$this covers the well-known Kohn–Nirenberg, anti-Kohn–Nirenberg and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functionsτand here we investigate the corresponding calculus in the model case of${\open R}^n$. We also give examples of nonlinearτappearing on the polarized and non-polarized Heisenberg groups.
Highlights
There are many ways of associating the operator to a function of variables (x, ξ) on the phase space
We are going to develop a calculus for (1.1), i.e. we prove the adjoint, composition and other formulae, make links between quantizations for different choices of τ, and investigate different properties of operators of this kind
In the case of G = Rn the quantization (1.4) reduces to (1.1), and since both mappings exp and log are identities, formula (1.5) boils down to τ (x) = (1/2)x, yielding the usual Weyl quantization, so that real-valued symbols are quantized into self-adjoint operators
Summary
There are many ways (quantizations) of associating the operator to a function of variables (x, ξ) on the phase space. If G is a locally compact unimodular group of type I, and τ : G → G is a measurable function, general τ -quantizations on G were considered in [6] in the form. In the case of G = Rn the quantization (1.4) reduces to (1.1), and since both mappings exp and log are identities, formula (1.5) boils down (modulo signs) to τ (x) = (1/2)x, yielding the usual Weyl quantization, so that real-valued (self-adjoint in the noncommutative case) symbols are quantized into self-adjoint operators. We get the formula for the ‘midpoint’ function m(x, y) = xτ (y−1x)−1 from (A.2) in the Weyl-type quantization (1.4) as m((a1, b1, c1), (a2, b2, c2)) = Such a midpoint between x and x−1 is not the origin but m((a, b, c), (a, b, c)−1) = 0, 0, − 2ab . The authors would like to thank Julio Delgado for discussions and for comments on the preliminary version of the manuscript
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
