Semi-classical pseudo-differential operators (Q2793611)

This chapter surveys the construction and properties of semi-classical pseudo-differential operators. The action of these operators is first described on Schwarz functions, and on \(L^2\) functions via operators bounds. The behavior of semi-classical pseudo-differential operators is then studied via asymptotic expansions under perturbations of their kernel outside of the diagonal. The Egorov theorem, which describes the conjugation of a semi-classical pseudo-differential operator by the Schrödinger propagator, is presented using symbolic calculus. The Gårding inequality is derived with its application to the positivity of weak limits of Wigner transforms, and asymptotic expansions are given for the functional calculus of operators. Finally, an extension of pseudo-differential operators to the setting of manifolds is considered, with a particular attention given to the case of the \(d\)-dimensional torus.NEWLINENEWLINEFor the entire collection see [Zbl 1310.37002].