Uncertainty Principle for Free Metaplectic Transformation

This study devotes to Heisenberg’s uncertainty inequalities of complex-valued functions in two free metaplectic transformation (FMT) domains without the assumption of orthogonality. In our latest work (Zhang in J Fourier Anal Appl 27(4):68, 2021), it is crucial that the FMT needs to be orthogonal for a decoupling of the cross terms. Instead of applying the orthogonality assumption, our current work uses the trace inequality for the product of symmetric matrices and positive semidefinite matrices to address the problem of coupling between cross terms. It formulates two types of lower bounds on the uncertainty product of complex-valued functions for two FMTs. The first one relies on the minimum eigenvalues of AjTAj-BjTAj,BjTAj,BjTBj-BjTAj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}_j^{\textrm{T}}\textbf{A}_j-\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{B}_j-\textbf{B}_j^{\textrm{T}}\textbf{A}_j$$\end{document}, while the other one relies on the minimum eigenvalues of AjTAj+BjTAj,BjTBj+BjTAj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}_j^{\textrm{T}}\textbf{A}_j+\textbf{B}_j^{\textrm{T}}\textbf{A}_j,\textbf{B}_j^{\textrm{T}}\textbf{B}_j+\textbf{B}_j^{\textrm{T}}\textbf{A}_j$$\end{document} and the maximum eigenvalues of BjTAj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_j^{\textrm{T}}\textbf{A}_j$$\end{document}, where Aj,Bj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}_j,\textbf{B}_j$$\end{document}, j=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} are the blocks found in symplectic matrices. Also, they are all relying on the covariance and absolute covariance. Sufficient conditions that truly give rise to the lower bounds are obtained. The theoretical results are verified by examples and experiments.