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arXiv · 2606.12964

Orthonormal Spectral Cluster Bounds on Manifolds with Nonpositive Curvature

Jean-Claude Cuenin·Ngoc Nhi Nguyen·Xiaoyan Su
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Abstract

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold with nonpositive sectional curvature. We prove sharp, logarithmically improved spectral cluster bounds for orthonormal systems in the supercritical range. More precisely, for spectral windows of size $(\log \lambda)^{-1}$, we obtain the orthonormal analogue of the logarithmically improved $L^q$ estimates of Hassell-Tacy. Our argument combines the universal orthonormal spectral cluster bounds of Frank-Sabin with B\'erard-type kernel estimates and a generalization of the Bourgain-Shao-Sogge-Yao multiplier estimate to the orthonormal setting.

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Jean-Claude Cuenin, Ngoc Nhi Nguyen, Xiaoyan Su. 2026-06-11. Orthonormal Spectral Cluster Bounds on Manifolds with Nonpositive Curvature. https://arxiv.org/abs/2606.12964

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