Matej Tusek: On thin quantum layers

Let $\Sigma$ be a surface in $R^3$ and $-\Delta_g$ the corresponding Laplace-Beltrami operator. Furthermore, define $\Omega_\epsilon$ as the layer of the width $\epsilon$ constructed around $\Sigma$. Then it is well known for some time that Laplace-Beltrami operator on $\Omega_\epsilon$ converges (after subtracting the divergent transverse energy term) to $-\Delta_{g}+K-M^2$ in the weak sense. Here $K$ and $M$ denote the Gauss and the mean curvature of $\Sigma$, respectively. In the first part of my talk, the norm resolvent convergence will be proved and consequently various convergences of the eigenfunctions will be discussed. In the second part, a brief overview of the nodal sets conjecture of Payne will be given and it will be supported by our result for thin quantum layers that follows from the above analysis. The talk is based on a joint work with David Krejcirik.