A symplectic extension map and a new Shubin class of pseudo-differential operators

Abstract

For an arbitrary pseudo-differential operator A:S(Rn)→S′(Rn) with Weyl symbol a∈S′(R2n), we consider the pseudo-differential operators A˜:S(Rn+k)→S′(Rn+k) associated with the Weyl symbols a˜=(a⊗12k)∘s, where 12k(x)=1 for all x∈R2k and s is a linear symplectomorphism of R2(n+k). We call the operators A˜ symplectic dimensional extensions of A. In this paper we study the relation between A and A˜ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of A˜ in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes HGρm1,m0 of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HGρm1,m0 but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.