Quantum harmonic analysis aims to establish operator-theoretic analogues of classical harmonic analysis results for spaces of operators (acting on Hilbert spaces). In this talk, we begin by recalling the existing notions of translation, modulation, and the Fourier transform of operators. We then discuss in detail a formulation of the Fourier transform of operators defined via “quantum Hermite functions”.
Building on this foundation, we introduce an alternative notion of translation and modulation of an operator using an irreducible representation of the twisted Heisenberg group, highlighting its analogy with the Euclidean setting. In particular, we establish quantum analogues of Young's inequality, Laplacian and the associated heat kernel. Finally, we explore a quantum analogue of modulation spaces constructed within this framework.
This talk is based on joint ongoing work with Prof. S. Thangavelu and Dr. Rahul Garg.
