This self-contained, graduate-level book presents a global pseudo-differential calculus in Euclidean spaces, which includes SG as well as Shubin classes and their natural generalizations containing Schroedinger operators with non-polynomial potentials.

This book is devoted to the global pseudo-differential calculus on Euclidean spaces and its applications to geometry and mathematical physics, with emphasis on operators of linear and non-linear quantum physics and travelling waves equations.The pseudo-differential calculus presented here has an elementary character, being addressed to a large audience of scientists. It includes the standard classes with global homogeneous structures, the so-called G and gamma operators.Concerning results for the applications, a first main line is represented by spectral theory. Beside complex powers of operators and asymptotics for the counting function, particular attention is here devoted to the non-commutative residue in Euclidean spaces and the Dixmier trace.Second main line is the self-contained presentation, for the first time in a text-book form, of the problem of the holomorphic extension of the solutions of the semi-linear globally elliptic equations. Entire extensions are discussed in detail. Exponential decay is simultaneously studied.

Addresses global pseudo-differential calculus and hypoellipticity Includes classic calculus as a special case Gives several applications to global problems in Euclidean spaces, including non-linear PDEs

Autorentext

Fabio Nicola is Full Professor of Mathematical Analysis at Politecnico di Torino, Italy. He has authored about 100 research articles and a monograph on several topics in partial differential equations, operator theory and Fourier analysis. S. Ivan Trapasso is a postdoctoral research fellow at the University of Genova, Italy. His main research interests are in the area of modern Fourier analysis, with applications to problems in mathematical physics and machine learning.

Zusammenfassung

This book presents a global pseudo-differential calculus in Euclidean spaces, which includes SG as well as Shubin classes and their natural generalizations containing Schroedinger operators with non-polynomial potentials. This calculus is applied to study global hypoellipticity for several pseudo-differential operators. The book includes classic calculus as a special case. It will be accessible to graduate students and of benefit to researchers in PDEs and mathematical physics.

Inhalt
Background meterial.- Global Pseudo-Differential Calculus.- ?-Pseudo-Differential Operators and H-Polynomials.- G-Pseudo-Differential Operators.- Spectral Theory.- Non-Commutative Residue and Dixmier Trace.- Exponential Decay and Holomorphic Extension of Solutions.