Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

In the present paper, we obtain some Fredholmness criteria and a formula for the index of nonlocal pseudodifferential operators generated by shifts and multiplications by periodic functions and acting in the Schwartz space S(ℝ n ). These results significantly differ from the results known to the author in this field of study, namely, the Fredholmness criteria and the formula for the index of nonlocal pseudodifferential operators acting over the noncompact manifold ℝ n are obtained here for the first time.