The index formula for
elliptic pseudodifferential operators on a two-dimensional manifold with conical
points contains the Atiyah-Singer integral as well as two additional terms. One of the
two is the ‘eta’ invariant defined by the conormal symbol, and the other term is
explicitly expressed via the principal and subprincipal symbols of the operator at
conical points. The aim of this paper is an explicit description of the contribution of a
conical point for higher-order differential operators. We show that changing the origin
in the complex plane reduces the entire contribution of the conical point to
the shifted ‘eta’ invariant. In turn this latter is expressed in terms of the
monodromy matrix for an ordinary differential equation defined by the conormal
symbol.
