Magnetic Pseudodifferential Operators

Abstract

In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in $R^n$ under the influence of a variable magnetic field $B$. It incorporates phase factors defined by $B$ and reproduces the usual Weyl calculus for $B=0$. In the present article we develop the classical pseum dodifferential theory of this formalism for the standard symbol classes $S_{\rho ,\delta }^m$. Among others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, properties of the magnetic Sobolev spaces. In the case when the vector potential $A$ has all the derivatives of order $\ge 1$ bounded, we show that the resolvent and the fractional powers of an elliptic magnetic pseudodifferential operator are also pseudodifferential. As an application, we get a limiting absorption principle and detailed spectral results for self-adjoint operators of the form $H=h(Q,{\Pi }^A)$, where $h$ is an elliptic symbol, $Q$ denotes multiplication with the variables ${\Pi }^A=D-A$, $D$ is the operator of derivation and $A$ is the vector potential corresponding to a short-range magnetic field.