The 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, spectral properties and complex powers of hypoelliptic operators, Schatten-von Neumann classes, the Connes-Wodzicki residue in Euclidean spaces, exponential decay of eigenfunctions, regularity and decay of homoclinics for semilinear equations, travelling waves. The book is completely self-contained and accessible to graduate students.
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