Anomalous real spectra of non-Hermitian quantum graphs in a strong-coupling regime

A family of one-dimensional quantum systems described by certain weakly non-Hermitian Hamiltonians with real spectra is studied. The presence of a microscopic spatial defect (i.e. of a fundamental length scale θ > 0) is mimicked by the replacement of the real line of coordinates R by a non-tree graph. By construction, this graph only differs from R on a small interval of size O(θ). An unexpected return of the reality of the whole spectrum in an anomalous strong-coupling regime (i.e. the emergence of an isolated island of stability of the system) is then revealed, proved and tentatively attributed to the topological nontriviality of the graph.