Spectral and propagation results for magnetic Schrödinger operators; A C∗-algebraic framework

We study generalized magnetic Schrödinger operators of the form Hh(A,V)=h(ΠA)+V, where h is an elliptic symbol, ΠA=−i∇−A, with A a vector potential defining a variable magnetic field B, and V is a scalar potential. We are mainly interested in anisotropic functions B and V. The first step is to show that these operators are affiliated to suitable C∗-algebras of (magnetic) pseudodifferential operators. A study of the quotient of these C∗-algebras by the ideal of compact operators leads to formulae for the essential spectrum of Hh(A,V), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same C∗-algebras by other ideals give localization results on the functional calculus of the operators Hh(A,V), which can be interpreted as non-propagation properties of their unitary groups.