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.

我们在带有 Clifford 丛的紧致黎曼流形上引入了新型分数广义椭圆算子。该理论适用于定义明确的微分几何。 Connes-Moscovici 定理根据 zeta 函数的留数给出了维数谱，适用于存在多个极点的情况。我们讨论了余切丛上单位共球面上的标量场问题，并将相关的迪克斯米尔迹评估为沃齐基留数。人们观察到不同类型的椭圆算子的出现，包括平方反比算子、分数算子和高阶算子，这些算子在理论物理中的循环上同调和指数问题等各个领域都有实用价值。