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

Abstract

Let \(\Gamma \) be a finite compact connected metric graph and let \(A\in L^\infty (\Gamma )\) be a real magnetic potential. The magnetic Laplacian \(H_A\) with standard vertex conditions is defined by the closed quadratic form

$$ \mathfrak q_A[u]=\sum _{e}\int _e |(-\mathrm i\partial _x-A_e)u_e|^2\,\mathrm dx . $$

The frustration-index Cheeger framework of Lange, Liu, Peyerimhoff and Post is adapted to compact metric graphs. The principal result identifies the frustration index on every regular subdomain with 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

$$ \lambda _0(H_A)\ge \frac{1}{5}\,h_A(\Gamma )^2, $$

where \(\lambda _0(H_A)\) is the bottom of the spectrum. In particular, if \(\delta _\Gamma (\Phi )\) denotes the flux distance of the global cycle-flux vector from the integral flux lattice, then

$$ \lambda _0(H_A)\ge \frac{1}{5} \min \left\{ h_\textrm{pr}(\Gamma ),\frac{\delta _\Gamma (\Phi )}{|\Gamma |}\right\} ^2 . $$

The corresponding heat-semigroup and magnetic-energy decay estimates follow from the spectral theorem. An upper bound with geometric dependence is also given, together with an example excluding a graph-independent upper bound by a constant multiple of \(h_A(\Gamma )^2\). The lower-bound constant is not asserted to be sharp.