Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian

TL;DR

This paper establishes bounds for the normal derivatives of spectral clusters of the Dirichlet Laplacian, extending previous results for eigenfunctions to spectral clusters on manifolds without trapped geodesics.

Contribution

It provides new upper and lower bounds for spectral cluster derivatives, generalizing earlier eigenfunction results to spectral clusters on Riemannian manifolds.

Findings

Upper bounds hold universally for any Riemannian manifold.

Lower bounds are valid for manifolds without trapped geodesics.

Results extend Hassell and Tao's eigenfunction bounds to spectral clusters.

Abstract

In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=\chi_{\lambda}^s f$ of Dirichlet Laplacian $\Delta_M$, $$c_s \lambda\|u\|_{L^2(M)} \leq \| \partial_{\nu}u \|_{L^2(\partial M)} \leq C_s \lambda \|u\|_{L^2(M)} $$ where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small $0<s<s_M$, where $s_M$ depends on the manifold only, provided that $M$ has no trapped geodesics (see Theorem \ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.