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Index of Differential-Difference Operators on an Infinite Cylinder

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Abstract

Differential-difference operators are considered on an infinite cylinder. The objective of the paper is to present an index formula for the operators in question. We define the operator symbol as a triple consisting of an internal symbol and conormal symbols on plus and minus infinity. The conormal symbols are families of operators with a parameter and periodic coefficients. Our index formula contains three terms: the contribution of the internal symbol on the base manifold, expressed by an analog of the Atiyah–Singer integral, the contributions of the conormal symbols at infinity, described in terms of the \(\eta\)-invariant, and also the third term, which also depends on the conormal symbol. The result thus obtained generalizes the Fedosov–Schulze–Tarkhanov formula.

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Acknowledgments

The author is grateful to A.Yu. Savin for posing the problem and for the help in writing the paper.

Funding

The research was supported in part by Young Russian Mathematics award, as well as RFBR and DFG, project number 21-51-12006.

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

    K. N. Zhuikov

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  1. K. N. Zhuikov
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Correspondence to K. N. Zhuikov.

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Zhuikov, K.N. Index of Differential-Difference Operators on an Infinite Cylinder. Russ. J. Math. Phys. 29, 280–290 (2022). https://doi.org/10.1134/S1061920822020091

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