The quantisation of normal velocity does not concentrate on hypersurfaces

TL;DR

This paper investigates the behavior of the quantised normal velocity of Laplacian eigenfunctions in the semiclassical regime, showing it does not concentrate on hypersurfaces, thus extending previous restriction results.

Contribution

It generalizes restriction estimates of Neumann data to a semiclassical setting for the normal velocity operator, demonstrating non-concentration properties.

Findings

Normal velocity quantisation does not concentrate on hypersurfaces

Restriction estimates hold for semiclassical quasimodes

Extends previous boundary restriction results to interior hypersurfaces

Abstract

We seek to extend work by Christianson-Hassell-Toth \cite{CHT} on restrictions of Neumann data of Laplacian eigenfunctions to interior hypersurfaces to a general semiclassical setting. In the semiclassical regime the appropriate generalisation is to study the restrictions of the function $v=\nu(x,hD)u$ where $\nu(x,hD)$ is the operator defined by quantising the normal velocity observable. For the Laplacian $\nu(x,hD)=\frac{1}{2}hD_{\nu}$ where $\nu$ is the normal to the hypersurface. We find that $||\nu(x,hD)u||_{L^{2}(H)}\lesssim||u||_{L^{2}(M)}$ provided $u$ is an $O_{L^{2}}(h)$ quasimode of the semiclassical pseudodifferential operator $p(x,hD)$. This statement should be interpreted as a statement of non-concentration for the quantisation of normal velocity.