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Circular symmetry and the trace formula. (English) Zbl 0686.58040
Let M be a compact n-dimensional manifold without boundary, and let P be a positive first order elliptic self-adjoint classical pseudo- differential operator on \(L^ 2(M)\), the Hilbert space of half-densities on M. Denote by \(\lambda_ 0\leq \lambda_ 1\leq \lambda_ 2\leq..\). the eigenvalues of P. Assume that P admits the circle group \(S^ 1\) as a symmetry group. The irreducible representations of \(S^ 1\) in \(L^ 2(M)\) are indexed by \({\mathbb{Z}}\) and accordingly one has the decomposition \(L^ 2(M)=\oplus_{m\in {\mathbb{Z}}}H_ m\) where \(H_ m\) are invariant subspaces of P. For each \(m>0\), let \(\lambda_{m,1}\leq \lambda_{m,2}\leq..\). be the eigenvalues of P restricted to \(H_ m\). The authors study the singularities of the distributions \(\sum_{j}e^{is\lambda_ j}\), \(\sum_{m,j}\phi (\lambda_{m,j}- Em)e^{ism}\), \(\sum_{m,j}\phi (\lambda_{m,j}- Em)e^{is\lambda_{m,j}}\) where \(\phi\in {\mathcal S}({\mathbb{R}})\) and \(E>0\). Applications to the trace formula for line bundles are presented.
Reviewer: P.Godin

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J70 Invariance and symmetry properties for PDEs on manifolds
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