Equilibrium and eigenfunctions estimates in the semi-classical regime

We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schr\"odinger operators, the asymptotic repartition of eigenvectors is the same as in the regular case, excepted in dimension 1 where a concentration at the critical point occurs. This principle extends to pseudo-differential operators and the limit measure is the Liouville measure as long as the singularity remains integrable.Comment: 13 pages, 1 figure, perhaps to be revise

Eigenfunction scarring and improvements in L∞L^{\infty} bounds

We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.Comment: 10 page

Averages of eigenfunctions over hypersurfaces

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ \phi_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $\mu$. Let $H$ be a smooth hypersurface. Our main result says that when $\mu$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.Comment: 18 pages, 1 figur

Explant Analysis of Total Disc Replacement

Explant analysis of human disc prostheses allow early evaluation of the host response to the prosthesis and the response of the prosthesis from the host. Furthermore, early predictions of failure and wear can be obtained. Thus far, about 2-3% of disc prostheses have been removed. Observed wear patterns are similar to that of appendicular prostheses including abrasions/scratching, burnishing, surface deformation, fatigue, and embedded debris. Chemically the polymeric components have shown little degradation in short-term implantation. In metal on metal prostheses the histologic responses consist of large numbers of metallic particles with occasional macrophages and giant cells. Only rare cases of significant inflammatory response from polymeric debris have been seen