On the index formula for singular surfaces

The index formula for elliptic pseudodifferential operators on a two-dimens ional manifold with conical points contains the Atiyah-Singer integral as w ell 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 a im of this paper is an explicit description of the contribution of a conica l 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 t erms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.