Quantum Cayley graphs for free groups

Abstract

Differential operators ∆+q are considered on metric Cayley graphs of the finitely generated free groups F M . The function q and the graph edge lengths may vary with the M edge types. Using novel methods, a set of M multipliers µ m (λ) depending on the spectral parameter is found. These multipliers are used to construct the resolvent and characterize the spectrum.2010 Mathematics Subject Classification 34B45, 58J50

Spectral estimates for infinite quantum graphs

“…For further results and references we refer to [56]. (iii) Using a completely different approach, the inequality λ 0 (H) > 0 was proved recently in [14,Theorem 4.16] for Cayley graphs of free groups under the additional symmetry assumption that edges in the same edge orbit have the same length. Since the graph G d is locally finite, τ is well defined on compactly supported functions and hence gives rise to a nonnegative symmetric pre-minimal operator in ℓ 2 (V; m).…”

Absolutely Continuous Spectrum for Quantum Trees

“…We are mainly interested in the case where the tree is the universal cover of some compact quantum graph. This implies the set of different lengths, potentials and coupling constants is finite, but the situation can be much more general than the special Cayley graph setting considered in [13]. We show in this framework that the spectrum will consist of (nontrivial) bands of pure AC spectrum, plus some discrete set of eigenvalues.…”

“…In case of regular trees T q , it was shown in [12] that the quantum tree obtained by endowing each edge with the same length L, the same symmetric potential W on the edges and the same coupling constant α at the vertices, has a spectrum consisting of bands of pure AC spectrum, along with eigenvalues between the bands. The setting was a bit generalized quite recently in [13], where each vertex in a 2q-regular tree is surrounded by the same set of lengths (L 1 , . .…”

“…However, such unimodular trees are still very general, they are actually the most interesting for us, and many techniques (such as a reduction to a half-line model) fail to tackle them. Even in the very simple case where the base graph G is regular but the edge lengths are not equal, the lifted structure in general will be neither radial periodic, nor identical around each vertex (in contrast to [13]).…”

“…In particular, the Cayley tree considered in [13] can be realized as the universal cover of the complete bipartite graph K 2q,2q , which is Hamiltonian. For this, use the fact that K 2q,2q has a proper 2q-edge-colouring and put the same length/potential on edges of the same colour.…”