Quantum star-graph analogues of PT-symmetric square wells

Abstract

We pick up a solvable $\mathcal{P}\mathcal{T}-$symmetric quantum square well on an interval of $x\in :=(-L,L){\mathbb{G}}^{(2)}$ (with an $\alpha -$dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval ${\mathbb{G}}^{(2)}$ (reinterpreted as an equilateral two-pointed star graph with the Kirchhoff matching at the vertex $x=0$) by a $q-$pointed equilateral star graph ${\mathbb{G}}^{(q)}$ endowed with the simplest complex-rotation-symmetric external $\alpha -$dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (1) at any integer $q=2,3,...$, there exists the same, $q-$independent and infinite subfamily of the real energies, and (2) at any special $q=2,6,10,...$, there exists another, additional and $q-$dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.

20 pp, 1 figure

References in corpus (30)

• Making Sense of Non-Hermitian Hamiltonians
• Supersymmetry and Quantum Mechanics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Observation of PT phase transition in a simple mechanical system
• SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra
• An infinite family of solvable and integrable quantum systems on a plane
• Self-isospectrality, special supersymmetry, and their effect on the band structure
• On the metric operator for the imaginary cubic oscillator
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Quantum Mechanics of Klein-Gordon-Type Fields and Quantum Cosmology
• Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system
• Non-Hermitian supersymmetry and singular PT symmetrized oscillators
• Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry
• PT-symmetric square well and the associated SUSY hierarchies
• Pseudo-Hermitian Quantum Mechanics with Unbounded Metric Operators
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Perfect transmission scattering as a PT-symmetric spectral problem
• Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
• PT-symmetric models in curved manifolds
• A Positive-Definite Scalar Product for Free Proca Particle
• PT-symmetric deformations of Calogero models
• Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Trigonometric identities, angular Schrödinger equations and a new family of solvable models
• Potential-controlled filtering in quantum star graphs
• N=2 supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane
• Complex Trajectories in a Classical Periodic Potential
• Anomalous real spectra of non-Hermitian quantum graphs in strong-coupling regime

Cited by in corpus (10)

• Non-self-adjoint graphs
• PT-symmetric quantum graphs
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Quantum Graphs: $\mathcal{P}\mathcal{T}$-symmetry and reflection symmetry of the spectrum
• Symmetrized exponential oscillator
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• Quantum dynamics of PT-symmetrically kicked particle confined in a 1D box
• Bound states without potentials: localization at singularities
• New Concept of Solvability in Quantum Mechanics