Let A be a A -algebra with real rank zero. Let A and A be Hilbert A -modules with A being full. Suppose that A is a linear map preserving orthogonality, ie,

In this paper, we generalize the anomaly cancellation formulas given by Alvarez-Gaum and Witten (1983), Liu (1995) and Han and Zhang (2004)[1],[2],[7] to the cases where an auxiliary bundle W and a complex line bundle are involved with no conditions on the first Pontryagin forms being assumed.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.

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