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 d
etail. Exponential decay is simultaneously studied.
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.
