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