Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension (Q2436381)

The authors propose a new quantization procedure for symbols \(\sigma(x,\eta)\) globally defined in \(\mathbb{R}^n\times \mathbb{R}^n\) satisfying the estimates \[ |\partial^\alpha_x \partial^\beta_\eta \sigma(x,\eta)|\leq C_{\alpha\beta}(1+|x|)^{m_1-|\alpha|}(1+ |\eta|)^{m_2-|\beta|}. \] Namely, an injective map \(F: S(\mathbb{R}^n)\to C^\infty(\mathbb{R}^{2n})\) is defined, by setting \[ Ff(v, t)= \sum_{u\in\mathbb{Z}^n} f(v+ u)\,e^{2\pi iut}. \] Let us denote by \(M(\mathbb{R}^{2n})\) the image \(F(S(\mathbb{R}^n))\). Starting from the classical quantization \(\sigma(x,D)\), the new quantization is defined as the operator \[ A_{\sigma}=F\sigma(x,D)F^{-1}:M(\mathbb{R}^{2n})\to M(\mathbb{R}^{2n}). \] Some applications are given concerning Fredholm operators, in particular index \(A_\sigma= \text{index}\,\sigma(x,D)\) is computed.