On elliptic differential operators with shifts

We give an index formula for elliptic differential operators whose coefficients include shifts forming an infinite group.

💡 Research Summary

The paper addresses a gap in index theory for elliptic differential operators whose coefficients contain shift operators generated by an infinite group. Classical Atiyah‑Singer index formulas apply to operators with coefficients that are smooth functions or finite‑dimensional group representations, but they break down when an infinite, possibly non‑abelian, group acts by translations (shifts) on the underlying manifold. The authors formulate a rigorous framework that incorporates such shifts and derive an explicit index formula that extends the traditional topological expression by adding a term reflecting the group’s cohomology.

The work begins by defining the shift operator T associated with a group element g∈G acting on a smooth manifold M: (Tf)(x)=f(g·x). When G is infinite (for example ℤ or a countable non‑abelian group), the combined operator D+T, where D is a standard elliptic differential operator, remains elliptic because its principal symbol σ(D+T)=σ(D)+σ(T) retains invertibility on the cotangent sphere bundle S* M. The authors prove ellipticity by showing that σ(T) is a bounded matrix‑valued function arising from a finite‑dimensional representation of G, and thus does not destroy the invertibility of σ(D).

To treat the non‑commutative nature of the group action, the paper embeds the problem into the crossed‑product C∗‑algebra A=C(M)⋊G. The operator D+T becomes an A‑module map, and its symbol defines a class