Mathematical Physics and Harmonic Analysis Seminar

Date: December 8, 2022

Time: 11:00AM - 11:50AM

Location: BLOC 302

Speaker: Alexander Kiselev

Title: Convergence of Neumann Laplacians on thin structures: an alternative approach

Abstract: Neumann Laplacians $A_\epsilon$ on thin manifolds, converging to metric graphs $G$ as $\epsilon\to0$, have been intensively studied by many authors, including Kuchment, Post, Pavlov, Exner, Zeng, among many others. The present-day state-of-the-art in this area is described in the monograph by O. Post.

It was proved that the spectra of $A_\epsilon$ converge within any compact $K\in \mathbb{C}$ in the sense of Hausdorff to the spectrum of a graph Laplacian $A_G$. In the book of Post, the claimed convergence was enhanced to the norm-resolvent type, with an explicit control of the error as $O(\epsilon^\gamma),$ with $\gamma>0$ explicitly given. The matching conditions at the vertices of the limiting graph turn out to be either:
(i) Kirchhoff (i.e. standard), if the vertex volumes are decaying, as $\epsilon\to0,$ faster than the edge volumes;
(ii) Resonant, which can be equivalently described in terms of $\delta$-type matching conditions with coupling constants proportional to the spectral parameter $z$, if the vertex and edge volumes are of the same order;
(iii) ``Dirichlet-decoupled" conditions (i.e., the graph Laplacian becomes completely decoupled), if the vertex volumes vanish slower than the edge ones.

In the talk, I will be primarily interested in the most non-trivial resonant case (ii). I will provide a straightforward, alternative to that of Post, proof of the fact that the Neumann Laplacians $A_\epsilon$ in this case converge in norm-resolvent sense to an ODE acting in the Hilbert space $L^2(G)\oplus \mathbb{C}^N$, where $N$ is the number of vertices. The operator to which it converges is in fact the one first pointed out by Kuchment as the self-adjoint operator whose spectrum coincides with the Hausdorff limit of spectra for the family $A_\e$.

I will show how a better error bound than that of Post is attained, namely, our estimate in the planar case is logarithmically worse than $O(\epsilon)$ and in the case of $\mathbb{R}^3$ is $O(\epsilo