Spectral and Propagation Results for Magnetic Schroedinger Operators; a C*-Algebraic Framework

Abstract

We study generalised magnetic Schroedinger operators of the form H(A,V)=h(P^A)+V, where h is an elliptic symbol, P^A is the generator of the magnetic translations, 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 H(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 H(A,V), which can be interpreted as non-propagation properties of their unitary groups.