Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs

TL;DR

This paper introduces non-Hermitian quantum graphs with real spectra, demonstrating their hidden Hermiticity through various representations, and explores their spectral properties and probabilistic interpretations.

Contribution

It presents a new class of non-Hermitian quantum graphs with real spectra, analyzed via discretization and different Hilbert space representations, revealing hidden Hermiticity and tunable nonlocality.

Findings

Spectra are real within a coupling interval independent of discretization.

Existence of hidden Hermiticity in different Hilbert space representations.

Explicit hermitizing inner products are provided in closed form.

Abstract

Manifestly non-Hermitian quantum graphs with real spectra are introduced and shown tractable as a new class of phenomenological models with several appealing descriptive properties. For illustrative purposes, just equilateral star-graphs are considered here in detail, with non-Hermiticities introduced by interactions attached to the vertices. The facilitated feasibility of the analysis of their spectra is achieved via their systematic approximative Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting bound-state spectra are found real in a discretization-independent interval of couplings. This conclusion is reinterpreted as the existence of a hidden Hermiticity of our models, i.e., as the standard and manifest Hermiticity of the underlying Hamiltonian in one of less usual, {\em ad hoc} representations ${\cal H}_j$ of the Hilbert space of states in which the inner product is local (at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples of these (of course, Hamiltonian-dependent) hermitizing inner products are offered in closed form. In this way each initial quantum graph is assigned a menu of optional, non-equivalent standard probabilistic interpretations exhibiting a controlled, tunable nonlocality.