Bounds for spectral projectors on tori

TL;DR

This paper studies the behavior of spectral projectors on tori, exploring their norms and related eigenfunction properties, proposing a conjecture and providing partial proofs to advance understanding in spectral analysis.

Contribution

It introduces a new conjecture about spectral projectors on tori and offers partial proofs, connecting resolvent bounds and eigenfunction norms.

Findings

Formulated a conjecture on spectral projectors on tori.

Provided partial proofs supporting the conjecture.

Linked spectral projector bounds to eigenfunction norm boundedness.

Abstract

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian, and to the boundedness of $L^p$ norms of eigenfunctions of the Laplacian. We formulate a conjecture, and partially prove it.