Adiabatic Limits and Related Lattice Point Problems

Abstract

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).