On the index of elliptic operators associated with groups of shifts

In this work, non-local elliptic operators on non-compact spaces are studied in the following situations. Firstly, differential-difference operators are considered on an infinite cylinder. The symbol of such operators consists of three components – the internal symbol and the conormal symbols at plus and minus infinity. The latter are families of parameter-dependent differential operators with periodic coefficients, for which the concept of the eta-invariant is introduced, and its main properties are proved. We obtain an index formula containing three terms - an analog of the Atiyah-Singer integral, the difference of eta-invariants at plus and minus infinity, and the third term, like the eta-invariant, depending on the conormal symbol. Secondly, we consider pseudodifferential operators on the real line with coefficients periodic at infinity. For differential operators, an index formula is obtained in terms of the monodromy matrices of limit operators at infinity, and the eta invariant is expressed in terms of the spectrum of the corresponding monodromy matrices. Thirdly, non-local operators in R^N associated with the metaplectic group are considered. A finiteness theorem is proved and explicit ellipticity conditions are found that guarantee the Fredholm property depending on the smoothness exponent of the Sobolev spaces in which the operator acts.
The results were partly obtained in joint work with A.Yu. Savin and P.A. Sipailo.