This article is cited in 3 papers

$\eta$-Invariant and index for operators on the real line periodic at infinity

A. Yu. Savin^ab, K. N. Zhuikov<sup>a

a</sup> S. M. Nikol'skii Mathematical Institute, Peoples' Friendship University of Russia (RUDN University), 6 Miklukho Maklaya St, 117198 Moscow, Russian Federation
^b Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

Abstract: We define $\eta$-invariants for periodic pseudodifferential operators on the real line and establish their main properties. In particular, it is proved that the $\eta$-invariant satisfies logarithmic property and a formula for the derivative of the $\eta$-invariant of an operator family with respect to the parameter is obtained. Furthermore, we establish an index formula for elliptic pseudodifferential operators on the real line periodic at infinity. The contribution of infinity to the index formula is given by the constructed $\eta$-invariant. Finally, we compute $\eta$-invariants of differential operators in terms of the spectrum of their monodromy matrices.

Keywords and phrases: elliptic operator, operator with periodic coefficients, $\eta$-invariant, index.

MSC: 58J20, 58J28, 58J40

Received: 31.07.2021

Language: English

DOI: 10.32523/2077-9879-2021-12-3-57-77

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