Reduction of the Calculus of Pseudodifferential Operators on a Noncompact Manifold to the Calculus on a Compact Manifold of Doubled Dimension

The paper is devoted to the exposition of results announced in [1] We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space $\mathbb{R}^{n}$ to a similar calculus in the space of sections of a one-dimensional fiber bundle $\xi$ on the $2n$-dimensional torus $\mathbb{T}^{2n}$. This reduction enables us to identify the Schwartz space on $\mathbb{R}^{n}$ with the space of smooth sections $\Gamma^{\infty}(T^{2n},\xi)$, compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of $\mathbb{R}^n$, we construct an index formula in accordance with the classical Atya–Singer formula.