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In this article we show how to compute the semiclassical spectral measure associated with the Schr¨odinger operator on Rn, and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schr¨odinger operator on R2 with a radially symmetric electric potential, V , and magnetic potential, B, both V and B are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schr¨odinger operator with its Birkhoff canonical form.

We construct a generalized Witten genus for spinc manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spinc manifolds called stringc manifolds. We also construct a mod 2 analogue of the Witten genus for 8k+2 dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalizedWitten genus and the mod 2 Witten genus on stringc and string (generalized) complete intersections in (product of) complex projective spaces respectively.

Given an open book decomposition (Σ,τ) of a three manifold Y , Thurston and Winkelnkemper [TW] construct a specific contact form a on Y . Given a spin-c Dirac operator D on Y , the contact form naturally associates a one parameter family of Dirac operators Dr = D− ir 2 cl(a) for r ≥ 0. When r >> 1, we prove that the spectrum of Dr = D0 − ir 2 cl(a) within [−1 2r 1 2 , 1 2r 1 2 ] are almost uniformly distributed. With the result in Part I [Ts1], it implies that the subleading order term of the spectral flow from D0 to Dr is of order r(logr) 9 2 . Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.

We give a proof of the Hard Lefschetz Theorem for orbifolds that does not involve intersection homology. We use a foliated version of the Hard Lefschetz Theorem due to El Kacimi.