Representation-theoretic framework for the linear canonical transform in quantum harmonic analysis

The Linear Canonical Transform (LCT) serves as a powerful generalization of the Fourier and fractional Fourier transforms, with significant implications in signal processing, optics, and quantummechanics. This paper develops a novel representation-theoretic framework for the LCT by leveraging the unitary dual of the Heisenberg group and the metaplectic representation of the symplectic group. Beyond recovering known uncertainty principles, we present refined inequalities that explicitly depend on the LCT parameter matrix and derive new structural results for the spectral decomposition of LCT operators. In particular, we provide a distributional spectral analysis for degenerate LCT cases (\(b = 0\)), introduce entropic uncertainty bounds tailored to the LCT domain, and propose a group-theoretic formulation of sparsity constraints. These findings significantly extend classical results and offer a deeper understanding of the LCT in both theoretical and applied contexts. We conclude with suggestions for quantum state manipulation via LCTs and numerical illustrations that bridge abstract theory with practical computation.