Fractional Elliptic Operators with Multiple Poles on Riemannian Manifold with Clifford Bundle

We introduce new types of fractional generalized elliptic operators on a compact Riemannian manifold with Clifford bundle. The theory is applicable in well-defined differential geometry. The Connes-Moscovici theorem gives rise to dimension spectrum in terms of residues of zeta functions, applicable in the presence of multiple poles. We have discussed the problem of scalar fields over the unit co-sphere on the cotangent bundle and we have evaluated the associated Dixmier traces as Wodzicki residues. It was observed the emergence of different types of elliptic operators, including inverse square, fractional and higher-order operators which are practical in various fields including cyclic cohomology and index problems in theoretical physics.