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The concept of the ????η-invariant was introduced in the famous work of Atiyah, Pathodi, and Singer for elliptic self-adjoint pseudodifferential operators (PDOs) on a closed manifold. It is by definition a spectral invariant, participates in many index formulas, and has many applications. An important generalization of the η-invariant was given by Melrose. Namely, it was proposed to consider the families ???? (????) of PDOs with a real parameter ????, and the η-invariant of the family was defined as a special regularization of the rotation number.

We will give the definition of the η-invariant for families of the form of linear combinations of PDOs with a parameter and periodic coefficients on a smooth closed manifold, and establish the basic properties of the ????-invariant. Then we will give the definition of the η-invariant of the PDO on the line with coefficients periodic at infinity.

The report will describe the results obtained in joint work with A.Yu. Savin.