Quantum Graphs: $\mathcal{P}\mathcal{T}$-symmetry and reflection symmetry of the spectrum

Abstract

Not necessarily self-adjoint quantum graphs -- differential operators on metric graphs -- are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $\mathcal{P}$. If the differential operator is $\mathcal{P}\mathcal{T}$-symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is $\mathcal{P}\mathcal{T}$-symmetric. We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is $\mathcal{P}\mathcal{T}$-symmetric with $\mathcal{P}$ being an automorphism of the metric graph.

arXiv comment16 pages, 1 figure

References in corpus (12)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• PT-Symmetric Quantum Mechanics
• Kirchhoff's Rule for Quantum Wires
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Can One Hear the Shape of a Graph?
• PT-symmetric quantum mechanics
• The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
• On the Spectral Gap of a Quantum Graph
• Non-self-adjoint graphs
• Point Interactions, Metamaterials, and PT-Symmetry
• Quantum star-graph analogues of PT-symmetric square wells
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra

Cited by in corpus (7)

• PT-symmetric quantum graphs
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Quantum graphs: self-adjoint, and yet exhibiting a nontrivial $\mathcal{P}\mathcal{T}$-symmetry
• Complex magnetic fields: An improved Hardy-Laptev-Weidl inequality and quasi-self-adjointness
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Kagome network with vertex coupling of a preferred orientation
• The Inverse Scattering Problem for the Matrix Schrödinger Equation