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

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