Periodic Jacobi matrices on trees need not have all gaps open]{Periodic Jacobi matrices on trees need not have all gaps open

A periodic Jacobi matrix on a tree is an operator on the universal cover of a finite connected graph. Its diagonal entries are lifted from a real potential on the graph's vertices, and its entries for adjacent vertices are lifted from positive weights on the edges. If the graph has $p$ vertices, the spectrum has at most $p-1$ gaps. Avni, Breuer and Simon conjectured that the potentials and weights for which all gaps are open, that is, for which the spectrum has exactly $p-1$ gaps, form a dense set. The conjecture fails for every connected $d$-regular graph with $d\ge3$ and $p\ge2$, except the graph with two vertices joined by $d$ edges, because the spectrum has fewer than $p-1$ gaps near the point where every weight is $1$ and the potential is $0$. More precisely, near that point, the spectrum of the operator is an interval if the graph is not bipartite, and it has at most one gap if the graph is bipartite. The proof uses the implicit function theorem.