Quantization of pseudo-differential operators on the torus

TL;DR

This paper studies pseudo-differential operators on the n-torus using Fourier series, establishing symbol correspondences, analyzing operator properties, and introducing Fourier series operators as analogues of Fourier integral operators.

Contribution

It introduces a global Fourier series approach to pseudo-differential operators on the torus, connecting toroidal and Euclidean symbols, and analyzes their boundedness and composition properties.

Findings

Established correspondence between toroidal and Euclidean symbols.

Derived composition formulas for Fourier series and pseudo-differential operators.

Proved boundedness of these operators on L^2 under certain conditions.

Abstract

Pseudo-differential and Fourier series operators on the n-torus are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on $L^2$ under certain conditions on their phases and amplitudes.