Magnetic Pseudodifferential Operators

In previous papers, a generalization of the Weyl calculus was introduced inconnection with the quantization of a particle moving in $\mathbb R^n$ underthe influence of a variable magnetic field $B$. It incorporates phase factorsdefined by $B$ and reproduces the usual Weyl calculus for B=0. In the presentarticle we develop the classical pseudodifferential theory of this formalismfor the standard symbol classes $S^m_{\rho,\delta}$. Among others, we obtainproperties and asymptotic developments for the magnetic symbol multiplication,existence of parametrices, boundedness and positivity results, properties ofthe magnetic Sobolev spaces. In the case when the vector potential $A$ has allthe derivatives of order $\ge 1$ bounded, we show that the resolvent and thefractional powers of an elliptic magnetic pseudodifferential operator are alsopseudodifferential. As an application, we get a limiting absorption principleand detailed spectral results for self-adjoint operators of the form$H=h(Q,\Pi^A)$, where $h$ is an elliptic symbol, $\Pi^A=D-A$ and $A$ is thevector potential corresponding to a short-range magnetic field