Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David Borthwick; Colin Guillarmou

Journal of the European Mathematical Society (2016)

  • Volume: 018, Issue: 5, page 997-1041
  • ISSN: 1435-9855

Abstract

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On geometrically finite hyperbolic manifolds Γ ∖ ℍ d , including those with non-maximal rank cusps, we give upper bounds on the number N ( R ) of resonances of the Laplacian in disks of size R as R → ∞ . In particular, if the parabolic subgroups of Γ satisfy a certain Diophantine condition, the bound is N ( R ) = 𝒪 ( R d ( log R ) d + 1 ) .

How to cite

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Borthwick, David, and Guillarmou, Colin. "Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds." Journal of the European Mathematical Society 018.5 (2016): 997-1041. <http://eudml.org/doc/277647>.

@article{Borthwick2016,
abstract = {On geometrically finite hyperbolic manifolds $\Gamma \backslash \mathbb \{H\}^\{d\}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R \rightarrow \infty $. In particular, if the parabolic subgroups of $\Gamma $ satisfy a certain Diophantine condition, the bound is $N(R)=\mathcal \{O\}(R^d (\mathrm \{log\} R)^\{d+1\})$.},
author = {Borthwick, David, Guillarmou, Colin},
journal = {Journal of the European Mathematical Society},
keywords = {spectral geometry; hyperbolic manifolds; resonances; spectral geometry; hyperbolic manifolds; resonances},
language = {eng},
number = {5},
pages = {997-1041},
publisher = {European Mathematical Society Publishing House},
title = {Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds},
url = {http://eudml.org/doc/277647},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Borthwick, David
AU - Guillarmou, Colin
TI - Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 5
SP - 997
EP - 1041
AB - On geometrically finite hyperbolic manifolds $\Gamma \backslash \mathbb {H}^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R \rightarrow \infty $. In particular, if the parabolic subgroups of $\Gamma $ satisfy a certain Diophantine condition, the bound is $N(R)=\mathcal {O}(R^d (\mathrm {log} R)^{d+1})$.
LA - eng
KW - spectral geometry; hyperbolic manifolds; resonances; spectral geometry; hyperbolic manifolds; resonances
UR - http://eudml.org/doc/277647
ER -

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