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Spectral theory of differential forms over a circle bundle

Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration : \begin{equation*} \mathbb{...
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introduction textbook to the Laplacian on a circle bundle

I am looking for an introduction to spectral theory of $\Delta$ on a circle bundle over a compact M. Is there an analog of Selberg trace formula?
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