Flux-explicit Cheeger bounds for magnetic Laplacians on compact metric graphs

Abstract

Let (Γ) be a finite compact connected metric graph and let (A\in L^\infty(Γ)) be a real magnetic potential. The magnetic Laplacian (H_A) with standard vertex conditions is defined by the closed quadratic form [ q_A[u]=\sum_e\int_e |(-i\partial_x-A_e)u_e|^2,dx. ] A magnetic Cheeger constant is introduced by adding to the usual boundary term the frustration index of the potential on subgraphs. The first point of the paper is that, on a metric graph, this frustration index is exactly a finite dimensional (\ell^1) flux distance determined by the periods of (A) on cycles. Consequently the Cheeger constant can be written directly in terms of Aharonov Bohm fluxes. We prove a Cheeger type lower bound for the bottom of the spectrum and derive the corresponding explicit lower estimate in terms of the distance of the global cycle flux vector from the integral flux lattice. The estimate also gives (L^2) decay bounds for the magnetic heat semigroup and for the magnetic energy. The constants are not asserted to be sharp; the emphasis is on the flux dependence and on the self contained metric graph formulation.