The pseudo-differential perturbations of the Dirac operator and the Kastler-Kalau-Walze type theorems

We define two types of pseudo-differential perturbations of the Dirac operator within the framework of the noncommutative geometry. And we obtain the noncommutative residue of the inverse square of these perturbations on 4-dimensional compact manifolds without boundary. As a generalization of the result of noncommutative residue on closed manifolds, we prove the Kastler-Kalau-Walze type theorems for these perturbations on 4-dimensional compact manifolds with boundary. Finally, several examples in which we can consider the corresponding pseudo-differential operators and the Kastler-Kalau-Walze type theorems are listed.