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

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.