Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

TL;DR

This paper extends spectral cluster bounds from scalar functions to orthonormal systems on Riemannian manifolds, using Schatten space bounds on oscillatory integral operators, and discusses their optimality.

Contribution

It generalizes Sogge's spectral cluster bounds to orthonormal systems and establishes Schatten bounds for oscillatory integral operators, highlighting their optimality.

Findings

Extended spectral cluster bounds to orthonormal systems.

Established Schatten bounds for oscillatory integral operators.

Discussed the optimality of the new bounds.

Abstract

We generalize the $L^p$ spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.