Adiabatic Limits and Related Lattice Point Problems

Conference paper
Part of the Trends in Mathematics book series (TM)


Let \((M,\mathcal{F})\) be a closed foliated manifold endowed with a Riemannian metric g. Then we have a direct sum decomposition \(TM\,=\,F \oplus H\) of the tangent bundle TM of M, where \(F\,=\,T\mathcal{F}\) is the tangent bundle of \(\mathcal{F}\) and \(H\,=\,{F}^{\perp }\) is the orthogonal complement of F, and the corresponding decomposition of the metric: \(g\,=\,{g}_{F} + {g}_{H}\). Consider the one-parameter family of Riemannian metrics on M,
$${g}_{\epsilon } = {g}_{F} + {\epsilon }^{-2}{g}_{ H},\quad \epsilon > 0,$$
and the corresponding Laplace-Beltrami operator \({\Delta }_{\epsilon }\). We are interested in the asymptotic behavior of the trace of the operator \(f({\Delta }_{\epsilon })\) for sufficiently nice functions f on \(\mathbb{R}\), in particular, of the eigenvalue distribution function \({N}_{\epsilon }(\lambda )\) of \({\Delta }_{\epsilon }\), as \(\epsilon \rightarrow 0\) (in the adiabatic limit).


Asymptotic Formula Orthogonal Complement Riemannian Metrics Adiabatic Limit Riemannian Foliation 
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Supported by the Russian Foundation of Basic Research (09-01-00389 and 12-01-00519).


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© Springer Basel 2013

Authors and Affiliations

  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia

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