The quantisation of normal velocity does not concentrate on 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.