Spectral singularities in PT-symmetric periodic finite-gap systems

Abstract

The origin of spectral singularities in finite-gap singly periodic PT-symmetric quantum systems is investigated. We show that they emerge from a limit of band-edge states in a doubly periodic finite gap system when the imaginary period tends to infinity. In this limit, the energy gaps are contracted and disappear, every pair of band states of the same periodicity at the edges of a gap coalesces and transforms into a singlet state in the continuum. As a result, these spectral singularities turn out to be analogous to those in the non-periodic systems, where they appear as zero-width resonances. Under the change of topology from a non-compact into a compact one, spectral singularities in the class of periodic systems we study are transformed into exceptional points. The specific degeneration related to the presence of finite number of spectral singularities and exceptional points is shown to be coherently reflected by a hidden, bosonized nonlinear supersymmetry.

arXiv comment16 pages, 3 figures; a difference between spectral singularities and exceptional points specified, the version to appear in PRD

References in corpus (61)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Exceptional Points of Non-hermitian Operators
• Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies
• Making the Case for Conformal Gravity
• Schrödinger operators with complex potential but real spectrum
• Encircling an Exceptional Point
• Invisibility in PT-symmetric complex crystals
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Spectral singularities and Bragg scattering in complex crystals
• Complex periodic potentials with real band spectra
• sl(2, C) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues
• Geometric phase around exceptional points
• Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions
• Observation of Exceptional Points in Electronic Circuits
• Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems
• Scalar Quantum Field Theory with Cubic Interaction
• A Twisted Kink Crystal in the Chiral Gross-Neveu model
• Self-isospectrality, special supersymmetry, and their effect on the band structure
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• Exceptional Points in a Microwave Billiard with Time-Reversal Invariance Violation
• Optical Spectral Singularities as Threshold Resonances
• Self-Isospectral Periodic Potentials and Supersymmetric Quantum Mechanics
• Exceptional Point Dynamics in Photonic Honeycomb Lattices with PT Symmetry
• Hidden supersymmetry in quantum bosonic systems
• Collectivity, Phase Transitions and Exceptional Points in Open Quantum Systems
• Spectral Singularities, Biorthonormal Systems, and a Two-Parameter Family of Complex Point Interactions
• Resonance Phenomenon Related to Spectral Singularities, Complex Barrier Potential, and Resonating Waveguides
• The Bogoliubov/de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry
• Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry
• Hidden nonlinear supersymmetries in pure parabosonic systems
• The energy spectrum of complex periodic potentials of the Kronig-Penney type
• Finite-gap systems, tri-supersymmetry and self-isospectrality
• On the shape of spectra for non-self-adjoint periodic Schrödinger operators
• An update on PT-symmetric complexified Scarf II potential, spectral singularities and some remarks on the rationally-extended supersymmetric partners
• Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit
• SUSY transformations between digonalizable and non-diagonalizable Hamiltonians
• On hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry
• Spectral singularities of non-Hermitian Hamiltonians and SUSY transformations
• Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry
• Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity
• Supersymmetric twisting of carbon nanotubes
• Numerical Simulations of PT-Symmetric Quantum Field Theories
• Is the CPT-norm always positive?
• On Darboux-Treibich-Verdier potentials
• Analytically Solvable PT-Invariant Periodic Potentials
• Perturbative Analysis of Spectral Singularities and Their Optical Realizations
• Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity
• Scattering cross section resonance originating from a spectral singularity
• Complex Periodic Potentials with a Finite Number of Band Gaps
• SUSY transformations with complex factorization constants. Application to spectral singularities
• PT-Invariant Periodic Potentials with a Finite Number of Band Gaps
• Irreducible second order SUSY transformations between real and complex potentials
• PT symmetry in relativistic quantum mechanics
• Peculiarities of the hidden nonlinear supersymmetry of Poschl-Teller system in the light of Lame equation
• Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity
• Astrophysical Evidence for the Non-Hermitian but $PT$-symmetric Hamiltonian of Conformal Gravity
• A complex periodic QES potential and exceptional points
• Spectral Singularities and Zero Energy Bound States
• New QES Hermitian as well as non-Hermitian PT invariant Potentials

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• Nonlinear Spectral Singularities for Localized Nonlinearities
• $PT$-symmetric invisible defects and confluent Darboux-Crum transformations
• Convective and absolute PT symmetry breaking in tight-binding lattices
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• Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
• Spectral Singularities and Whispering Gallery Modes of a Cylindrical Gain Medium
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• Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials
• Lasing Threshold Condition for Oblique TE and TM Modes, Spectral Singularities, and Coherent Perfect Absorption
• Spectral Singularities in the Surface Modes of a Spherical Gain Medium
• Wave emission and absorption at spectral singularities
• Non-Hermitian interferometer: Unidirectional amplification without distortion
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• Twisted kinks, Dirac transparent systems and Darboux transformations
• Spectral Singularities in the TE and TM modes of a PT-Symmetric Slab System: Optimal conditions for realizing a CPA-Laser
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• Talbot self-imaging in $\mathcal{P}\mathcal{T}$-symmetric complex crystals
• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• Spectral Singularity in confined PT symmetric optical potential
• Perturbative Unidirectional Invisibility
• Nonlinear Spectral Singularities and Laser Output Intensity
• Conserved Correlation in PT -symmetric Systems: Scattering and Bound States
• Excepional Points from Hamiltonians of hybrid physical systems: Squeezing and anti-Squeezing
• Complete particle-pair annihilation as a dynamical signature of the spectral singularity
• Transmission phase lapse in the non-Hermitian Aharonov-Bohm interferometer near the spectral singularity
• Spectral singularities and tunable slab lasers with 2D material coating
• Exceptional point in a coupled Swanson system
• Pöschl-Teller Hamiltonian: Gazeau-Klauder type coherent states, related statistics and geometry
• Exceptional points and quantum phase transition in a fermionic extension of the Swanson oscillator
• Solving scattering problems in the half-line using methods developed for scattering in the full line
• Nonlinear supersymmetry as a hidden symmetry
• Analogue Hawking radiation as a tunneling in a two-level $\mathcal{P}\mathcal{T}$-symmetric system
• A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
• Spectral Singularities, Threshold Gain, and Output Intensity for a Slab Laser with Mirrors
• Applications of the potential algebras of the two-dimensional Dirac-like operators
• Pseudo-Hermitian extensions of the harmonic and isotonic oscillators
• Double-sided unidirectional reflectivity
• Artificial Hawking radiation, weak pseudo-Hermiticity and Weyl semimetal blackhole analogy
• Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model
• Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{P}\mathcal{T}$-Symmetric $(-x^4)$ Theory
• PT-symmetric transport in non-PT-symmetric bi-layer optical arrays
• Quasi-isospectral higher-order Hamiltonians via a reversed Lax pair construction
• Algebraic integrability of $\mathcal{P}\mathcal{T}$-deformed Calogero models
• Swanson Hamiltonian: non-PT-symmetry phase
• Spectral singularities and threshold gain of a slab laser under illumination of a focused Gaussian beam
• Free Particle to Complex KdV breathers through Isospectral Deformation
• Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
• Non-Hermitian superintegrable systems