Convergence of the Neumann Laplacian on Open Book Structures
Abstract
We consider a compact C ∞-stratified 2D variety M in R^3 and its ε-neighborhood Mvε , which we call a “fattened open book structure.” Assuming absence of zero-dimensional strata, i.e. “corners,” we show that the (discrete) spectrum of the Neumann Laplacian in M, converges when ε→ 0 to the spectrum of a differential operator on M.
Similar results have been obtained before for the case of fattened graphs, i.e. M being one-dimensional. In the case of a 2D smooth submanifold M, the problem has been studied well. However, having singularities along strata of lower dimensions significantly complicates considerations. As in the quantum graph case, such considerations are triggered by various applications such as micro-electronics, photonic devices, and dynamical systems with two “slow” and one “fast” degrees of freedom. The results are obtained under two restrictions: 1) there are no zero dimensional strata (corners); 2) the pages are transverse at the bindings (no cusps). We begin with the “uniformly fattened case:” width of the fattened domain shrinks with the same speed around “pages” and “bindings.” Next we consider more general fattened open book structures with a finite number of parameters which control the size of the fattened neighborhood around each point. In particular we consider β -sized neighborhoods around the bindings and ε-sized neighborhood around the pages. By properly tuning these parameters, we demonstrate three classes of limit operators on M. We show that there is a relative length scale (controlled by β) between the “fattened pages” and “fattened binding” which causes the system to undergo phase transitions. Two such phases have novel boundary currents along the bindings.
Citation
Corbin, James Edward (2019). Convergence of the Neumann Laplacian on Open Book Structures. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /188720.