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

For an arbitrary pseudo-differential operator A : S(R-n) -> S '(R-n) with Weyl symbol a is an element of S '(R-2n), we consider the pseudo-differential operators (A) over tilde : S(Rn+k) -> S '(Rn+k) associated with the Weyl symbols (a) over tilde = (a circle times 1(2k)) o s, where 1(2k) (x) = 1 for all x is an element of R-2k and s is a linear symplectomorphism of R2(n+k). We call the operators (A) over tilde symplectic dimensional extensions of A. In this paper we study the relation between A and (A) over tilde in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of (A) over tilde 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 HG(rho)(m1,m0) of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in HG rho(m1,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. (C) 2013 Elsevier Inc. All rights reserved.