di?erential operators in particular will be developed hand in glove with appli- tions andcomputation inthe physical,biologicaland medicalsciences.This theme will play an important role in the forthcoming volumes on pseudo-di?erential - erators originating from IGPDO. The Editors OperatorTheory: Advances andApplications,Vol.189, 1-14 c 2008Birkh. auserVerlagBasel/Switzerland Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Di?erential Operators Maurice de Gosson Abstract. In a recent series of papers M.W. Wong has studied a degenerate elliptic partial di?erential operator related to the Heisenberg group. It turns out that Wong's example is best understood when replaced in the context of the phase-space Weyl calculus we have developed in previous work; this - proach highlights the relationship of Wong's constructions with the quantum mechanics of charged particles in a uniform magnetic ?eld. Using Shubin's classes of pseudodi?erential symbols we prove global hypoellipticity results for arbitrary phase-space operators arising from elliptic operators on con- uration space. Mathematics Subject Classi?cation (2000).Primary 47F30; Secondary 35B65, 46F05. Keywords. Degenerate elliptic operators, hypoellipticity, phase space Weyl calculus, Shubin symbols.

di?erential operators in particular will be developed hand in glove with appli- tions andcomputation inthe physical,biologicaland medicalsciences.This theme will play an important role in the forthcoming volumes on pseudo-di?erential - erators originating from IGPDO. The Editors OperatorTheory: Advances andApplications,Vol.189, 1-14 c 2008Birkh. auserVerlagBasel/Switzerland Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Di?erential Operators Maurice de Gosson Abstract. In a recent series of papers M.W. Wong has studied a degenerate elliptic partial di?erential operator related to the Heisenberg group. It turns out that Wong's example is best understood when replaced in the context of the phase-space Weyl calculus we have developed in previous work; this - proach highlights the relationship of Wong's constructions with the quantum mechanics of charged particles in a uniform magnetic ?eld. Using Shubin's classes of pseudodi?erential symbols we prove global hypoellipticity results for arbitrary phase-space operators arising from elliptic operators on con- uration space. Mathematics Subject Classi?cation (2000).Primary 47F30; Secondary 35B65, 46F05. Keywords. Degenerate elliptic operators, hypoellipticity, phase space Weyl calculus, Shubin symbols.

di?erential operators in particular will be developed hand in glove with appli- tions andcomputation inthe physical,biologicaland medicalsciences.This theme will play an important role in the forthcoming volumes on pseudo-di?erential - erators originating from IGPDO. The Editors OperatorTheory: Advances andApplications,Vol.189, 1-14 c 2008Birkh. auserVerlagBasel/Switzerland Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Di?erential Operators Maurice de Gosson Abstract. In a recent series of papers M.W. Wong has studied a degenerate elliptic partial di?erential operator related to the Heisenberg group. It turns out that Wong's example is best understood when replaced in the context of the phase-space Weyl calculus we have developed in previous work; this - proach highlights the relationship of Wong's constructions with the quantum mechanics of charged particles in a uniform magnetic ?eld. Using Shubin's classes of pseudodi?erential symbols we prove global hypoellipticity results for arbitrary phase-space operators arising from elliptic operators on con- uration space. Mathematics Subject Classi?cation (2000).Primary 47F30; Secondary 35B65, 46F05. Keywords. Degenerate elliptic operators, hypoellipticity, phase space Weyl calculus, Shubin symbols.