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 $\nu (x,hD)=\frac{1}{2}h{D}_{\nu }$ 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.

No change to major theorem from version 2. Some changes to technical results in Section 1. Final version

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