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

Abstract

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 ℝn to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus \(\mathbb{T}^{2n} \). This reduction enables us to identify the Schwartz space on ℝn with the space of smooth sections Γ∞(T2n, ξ), 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 ℝn, we construct an index formula in accordance with the classical Atya-Singer formula.