Norm-resolvent convergence for Neumann Laplacians on manifolds thinning to graphs

TL;DR

This paper proves sharp norm-resolvent convergence of Neumann Laplacians on thin domains to quantum graphs, clarifying the vertex conditions as related to delta-prime interactions, in the resonant case.

Contribution

It establishes order-sharp convergence results for Neumann Laplacians on thin domains to metric graphs, revealing the nature of vertex conditions in the limit.

Findings

Sharp norm-resolvent convergence with error estimates

Vertex conditions related to delta-prime interactions

Convergence in the resonant case

Abstract

Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^d,$ $d\ge2,$ converging to metric graphs in the limit of vanishing thickness parameter in the resonant case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to $\delta'$ type.