Quantization of Pseudo-differential Operators on the Torus

By Michael Ruzhansky, Ville T. Turunen. 2008

171 citations recorded by the source corpus.

AbstractPseudo-differential and Fourier series operators on the torus ${{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n}$ 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 L2 under certain conditions on their phases and amplitudes.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5f1693f045249f4abd73609ff33d11737fda4603

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In Applied Mathematics

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Citations in the corpus

Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces (2008, background): “If we denote by S m (T n × Z n ) the class of functions p : T n × Z n → C satisfying (1.3), then we have OpS m (T n × Z n ) = � m (T n ), see e.g. [1, 13, 33, 16, 17 ]”
Nonharmonic Analysis of Boundary Value Problems (2015, methodology): “The following example shows that among other things, we can extend to the nonself-adjoint setting the toroidal calculus (with periodic boundary conditions) developed in [54].”
Schatten classes and traces on compact groups (2013, background): “has been analysed extensively in [16] and it is a special case of (2.”
Quantization on Nilpotent Lie Groups (2016, methodology): “for a thorough analysis of which we refer to [RT10b] and [RT10a, Section 4].”
Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity (2010, background): “We refer to [11] for details of the corresponding toroidal quantization of operators on Tn .”