Pseudo-Hermitian Representation of Quantum Mechanics

Abstract

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.

76 pages, 2 figures, 243 references, published as Int. J. Geom. Meth. Mod. Phys. 7, 1191-1306 (2010)

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• The ODE/IM Correspondence
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Cited by in corpus (566)

• Non-Hermitian Physics
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
• Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial
• Biorthogonal Quantum Mechanics
• Edge states and topological phases in non-Hermitian systems
• Quantum state tomography across the exceptional point in a single dissipative qubit
• Nonlinear switching and solitons in PT-symmetric photonic systems
• Making the Case for Conformal Gravity
• Non-Hermitian systems and topology: A transfer-matrix perspective
• Minimal length, maximal momentum and Hilbert space representation of quantum mechanics
• Observation of PT-symmetric quantum interference
• Twofold Transition in PT-Symmetric Coupled Oscillators
• PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold
• Pseudospectra in non-Hermitian quantum mechanics
• Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry
• Anti-Parity-Time Symmetric Optical Four-Wave Mixing in Cold Atoms
• Non-Hermitian Hamiltonians and no-go theorems in quantum information
• On the metric operator for the imaginary cubic oscillator
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Non-Hermitian exceptional Landau quantization in electric circuits
• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Spectral Singularities, Biorthonormal Systems, and a Two-Parameter Family of Complex Point Interactions
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Entanglement Entropy of Non Unitary Conformal Field Theory
• Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric Potential
• Boundary-dependent Self-dualities, Winding Numbers and Asymmetrical Localization in non-Hermitian Quasicrystals
• The Theorem of Ostrogradsky
• Optical waveguide arrays: quantum effects and PT symmetry breaking
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• All-real spectra in optical systems with arbitrary gain and loss distributions
• Theory of superconductivity with non-Hermitian and parity-time reversal symmetric cooper pairing symmetry
• Classical Simulation of Relativistic Quantum Mechanics in Periodic Optical Structures
• Magnetic suppression of non-Hermitian skin effects
• Measuring Non-Hermitian Operators via Weak Values
• Parity-time symmetry in optical microcavity systems
• The non-Hermitian Swanson model with a time-dependent metric
• On the Quantisation of Complex Higher Derivative Theories and Avoiding the Ostrogradsky Ghost
• Analytic Results for a PT-symmetric Optical Structure
• A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field
• Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Time correlation functions for non-Hermitian quantum systems
• Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry
• Non-Hermitian skin effect as an impurity problem
• Eigenvalue structure of a Bose-Einstein condensate in a PT-symmetric double well
• Time-dependent Hamiltonians with 100% evolution speed efficiency
• $PT$-symmetric invisible defects and confluent Darboux-Crum transformations
• Non-Hermiticity and topological invariants of magnon Bogoliubov-de Gennes systems
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Gegenbauer-solvable quantum chain model
• Curving the space by non-Hermiticity
• Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Spectral Singularities of a Complex Spherical Barrier Potential and Their Optical Realization
• Transfer Matrices as Non-Unitary S-Matrices, Multimode Unidirectional Invisibility, and Perturbative Inverse Scattering
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Hamiltonian formulation of the standard $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger dimer
• Time-dependent q-deformed coherent states for generalized uncertainty relations
• On Symmetries and Exact Solutions of a Class of Non-local Non-linear Schrodinger Equations with Self-induced PT-symmetric Potential
• Dimer with gain and loss: Integrability and $\mathcal{P}\mathcal{T}$-symmetry restoration
• Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations
• Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory
• Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system
• Entanglement Entropy of Non-Unitary Integrable Quantum Field Theory
• Mending the broken PT-regime via an explicit time-dependent Dyson map
• Passive $\mathcal{P}\mathcal{T}$-symmetry breaking transitions without exceptional points in dissipative photonic systems
• Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
• Generating high-order quantum exceptional points in synthetic dimensions
• Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras
• Pseudo-Hermitian Quantum Mechanics with Unbounded Metric Operators
• Scattering theory using smeared non-Hermitian potentials
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• From scattering theory to complex wave dynamics in non-hermitian PT-symmetric resonators
• Pseudo-Hermitian approach to Goldstone's theorem in non-Abelian non-Hermitian quantum field theories
• Perfect transmission scattering as a PT-symmetric spectral problem
• Bohmian quantum trajectories from coherent states
• Observation of slowly decaying eigenmodes without exceptional points in Floquet, dissipative, synthetic circuits
• Eternal life of entropy in non-Hermitian quantum systems
• Simulating broken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems by weak measurement
• Conserved quantities in parity-time symmetric systems
• Linear Pseudo-fermions
• Semiclassical Analysis of Spectral Singularities and Their Applications in Optics
• Irreversibility of the renormalization group flow in non-unitary quantum field theory
• Non-Hermitian Hamiltonian for a Modulated Jaynes-Cummings Model with PT Symmetry
• Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator
• Spectral Singularities of a General Point Interaction
• PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability
• From self-adjoint to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls
• Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity
• Hermitian dynamics in a class of pseudo-Hermitian networks
• Momentum-independent reflectionless transmission in the non-Hermitian time-reversal symmetric system
• Non-self-adjoint hamiltonians defined by Riesz bases
• Interpretation of quantum mechanics with indefinite norm
• Restoring Adiabatic State Transfer in Time-Modulated Non-Hermitian Systems
• Hermitian versus non-Hermitian representations for minimal length uncertainty relations
• Constructible reality condition of pseudo entropy via pseudo-Hermiticity
• Emergent $\mathcal{P}\mathcal{T}$ symmetry in a double-quantum-dot circuit QED set-up
• More mathematics for pseudo-bosons
• Regularized degenerate multi-solitons
• Quantum information dynamics in a high-dimensional parity-time-symmetric system
• Quantum catastrophes: a case study
• Non-Hermitian skin effect edge
• Probability-preserving evolution in a non-Hermitian two-band model
• Investigation of non-Hermitian Hamiltonians in the Heisenberg picture
• Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics
• New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics
• PT-symmetry breaking in complex nonlinear wave equations and their deformations
• Unitarity corridors to exceptional points
• Directed momentum current induced by the PT-symmetric driving
• Solvable two dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime
• Eigenstate entanglement entropy in $PT$ invariant non-Hermitian system
• Non-hermitian operator modelling of basic cancer cell dynamics
• Non-self-adjoint graphs
• New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition
• Parametric symmetries in exactly solvable real and P T symmetric complex potentials
• Density Matrix Formalism for PT-Symmetric Non-Hermitian Hamiltonians with the Lindblad Equation
• The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
• Exactly solvable time-dependent pseudo-Hermitian $su$(1,1) Hamiltonian models
• Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry
• Massive gauge particles versus Goldstone bosons in non-Hermitian non-Abelian gauge theory
• Entanglement, non-Hermiticity, and duality
• 't Hooft-Polyakov monopoles in non-Hermitian quantum field theory
• Non-Hermitian Heisenberg representation
• Deformed quons and bi-coherent states
• On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
• Quantum Mechanics of a Photon
• Non-Hermitian dynamics of slowly-varying Hamiltonians
• Generic examples of PT-symmetric qubit (spin 1/2) Liouvillians
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• Dyson Maps and Unitary Evolution for Maxwell Equations in Tensor Dielectric Media
• Confluent Crum-Darboux transformations in Dirac Hamiltonians with $PT$-symmetric Bragg gratings
• A non self-adjoint model on a two dimensional noncommutative space with unbound metric
• Unidirectional Invisibility and PT-Symmetry with Graphene
• Maximal quantum entanglement at exceptional points via unitary and thermal dynamics
• Spectral Singularities in the TE and TM modes of a PT-Symmetric Slab System: Optimal conditions for realizing a CPA-Laser
• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
• Exact description of coalescing eigenstates in open quantum systems in terms of microscopic Hamiltonian dynamics
• Imaginary-Scaling versus Indefinite-Metric Quantization of the Pais-Uhlenbeck Oscillator
• Various Scattering Properties of a New PT-symmetric non-Hermitian potential
• Two-qubit entanglement generation through non-Hermitian Hamiltonians induced by repeated measurements on an ancilla
• Exactly solvable time-dependent non-Hermitian quantum systems from point transformations
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Deconfinement to Confinement as PT phase transition
• Crypto-unitary forms of quantum evolution operators
• Linear Quantum Entropy and Non-Hermitian Hamiltonians
• Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity
• Soliton defects in one-gap periodic system and exotic supersymmetry
• Transition Probabilities in the Two-Level Quantum System with PT-Symmetric Non-Hermitian Hamiltonians
• A no-go result for the quantum damped harmonic oscillator
• Degeneracies and symmetry breaking in pseudo-Hermitian matrices
• Degenerate multi-solitons in the sine-Gordon equation
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• On the spectral properties of the Schrodinger operator with a periodic PT-symmetric potential
• Minimal areas from q-deformed oscillator algebras
• Non-Hermitian impurities in Dirac systems
• A note on topological insulator phase in non-hermitian quantum systems
• Antilinear deformations of Coxeter groups, an application to Calogero models
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Asymmetric scattering by non-hermitian potentials
• Multiply Degenerate Exceptional Points and Quantum Phase Transitions
• Anomalous Parity-Time Symmetry Transition away from an Exceptional Point
• Some remarks on quasi-Hermitian operators
• From Pseudo-bosons to Pseudo-Hermiticity via multiple generalized Bogoliubov transformations
• Embedding, simulation and consistency of $\mathcal{P}\mathcal{T}$ -symmetric quantum Theory
• Partial inner product spaces, metric operators and generalized hermiticity
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Weak Ergodicity Breaking in Non-Hermitian Many-body Systems
• Three solvable matrix models of a quantum catastrophe
• Infinite families of (non)-Hermitian Hamiltonians associated with exceptional $X_m$ Jacobi polynomials
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Spectral singularity and non-Hermitian PT-symmetric extension of $A_{N-1}$ type Calogero model without confining potential
• Realization of complex conjugate media using non-PT-symmetric photonic crystals
• Exact solutions for time-dependent non-Hermitian oscillators: classical and quantum pictures
• PT phase transition in a (2+1)-d relativistic system
• Horizonless ultracompact objects and dark matter in quadratic gravity
• Photonic quantum simulations of coupled $PT$-symmetric Hamiltonians
• Time-delay and reality conditions for complex solitons
• Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach
• New scattering features in non-Hermitian space fractional quantum mechanics
• Passage through exceptional point: Case study
• Quantizations from reproducing kernel spaces
• Quantum phase transitions in nonhermitian harmonic oscillator
• Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems
• From real fields to complex Calogero particles
• Discrete quantum square well of the first kind
• Real spectra, Anderson localization, and topological phases in one-dimensional quasireciprocal systems
• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• Maxwell meets Reeh-Schlieder: the quantum mechanics of neutral bosons
• Pseudo-Hermitian random matrix theory
• Floquet exceptional contours in Lindblad dynamics with time-periodic drive and dissipation
• Complex symmetric Hamiltonians and exceptional points of order four and five
• Non linear pseudo-bosons versus hidden Hermiticity. II: The case of unbounded operators
• Exotic entanglement for non-Hermitian Jaynes-Cummings Hamiltonians
• Decays of degeneracies in PT-symmetric ring-shaped lattices
• Bicomplex Hamiltonian systems in Quantum Mechanics
• PT symmetry in relativistic quantum mechanics
• Pseudo-Hermitian Hamiltonians Generating Waveguide Mode Evolution
• Non-self-adjoint hamiltonians defined by generalized Riesz bases
• Extension of Noether's theorem in PT-symmetric systems and its experimental demonstration in an optical setup
• Nonlinear wave dynamics near phase transition in $\mathcal{P}\mathcal{T}$-symmetric localized potentials
• Analytically solvable $2\times 2$ $PT$-symmetry dynamics from su(1,1)-symmetry problems
• On-demand Parity-Time symmetry in a lone oscillator through complex, synthetic gauge fields
• Hidden PT Symmetry and quantization of coupled-oscillators model of QASER
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Spectral Singularities and CPA-Laser Action in a Weakly Nonlinear PT-Symmetric Bilayer Slab
• Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians
• Metric versus observable operator representation, higher spin models
• Non-Hermitian noncommutative quantum mechanics
• Scaling laws of the out-of-time-order correlators at the transition to the spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking in a Floquet system
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Competing PT potentials and re-entrant PT symmetric phase for a particle in a box
• Quantum star-graph analogues of PT-symmetric square wells
• E2-quasi-exact solvability for non-Hermitian models
• Dissipative dynamics in open XXZ Richardson-Gaudin models
• Non-Hermitian gauge field theories and BPS limits
• $\mathcal{P}\mathcal{T}$-symmetry breaking in a Kitaev chain with one pair of gain-loss potentials
• Time-independent approximations for time-dependent optical potentials
• Novel Effect Induced by Spacetime Curvature in Quantum Hydrodynamics
• Nonlinear symmetries of perfectly invisible $PT$-regularized conformal and superconformal mechanics systems
• Point Interactions, Metamaterials, and PT-Symmetry
• Non linear pseudo-bosons versus hidden Hermiticity
• PT-symmetry breaking in resonant tunneling heterostructures
• PT-symmetry breaking with divergent potentials: lattice and continuum cases
• General theory of regular biorthogonal pairs and its physical applications
• Chaotic diffusion of complex trajectory and its quantum signature
• The Floquet Baxterisation
• The Spectrum of the Hamiltonian with a PT-symmetric Periodic Optical Potential
• Balanced gain and loss in Bose-Einstein condensates without PT symmetry
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices
• Unbounded C-symmetries and their nonuniqueness
• Detecting exceptional points through dynamics in non-Hermitian systems
• Perturbation theory near degenerate exceptional points
• Milne quantization for non-Hermitian systems
• A black potential for spin less particles
• Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and $c\to \infty$ Limit
• Spontaneous PT-symmetry breaking for systems of noncommutative Euclidean Lie algebraic type
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator
• Holonomic Gates in Pseudo-Hermitian Quantum Systems
• The effect of thermal photons on exceptional points in coupled resonators
• Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions
• Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation
• Spectrum and normal modes of non-hermitian quadratic boson operators
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• $Sp(4;\mathbb{R})$ Squeezing for Bloch Four-Hyperboloid via The Non-Compact Hopf Map
• Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy
• Exactly solvable $\mathcal{P}\mathcal{T}$-symmetric models in two dimensions
• Observables in non-Hermitian systems: A methodological comparison
• Observing a Changing Hilbert-Space Inner Product
• Theory of response to perturbations in non-Hermitian systems using five-Hilbert-space reformulation of unitary quantum mechanics
• Complex BPS solitons with real energies from duality
• Arnold's potentials and quantum catastrophes
• From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier
• Entangled Quantum State Discrimination using Pseudo-Hermitian System
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Analysis of half-spin particle motion in Kerr-Newman field by means of effective potentials in second-order equations
• Pseudo-invariant approach for a particle in a complex time-dependent linear potential
• Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility
• Gyrating solitons in a necklace of optical waveguides
• Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra
• No-signaling principle and Bell inequality in PT-symmetric quantum mechanics
• Magnonic analog of relativistic Zitterbewegung in an antiferromagnetic spin chain
• Anomalous Faraday effect in a $\mathcal{P}\mathcal{T}$-symmetric dielectric slab
• Composite quantum Coriolis forces
• On the K-theoretic classification of dynamically stable systems
• Formation of Exceptional Points in pseudo-Hermitian Systems
• Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models
• Hilbert space inner products for PT-symmetric Su-Schrieffer-Heeger models
• A chain of solvable non-Hermitian Hamiltonians constructed by a series of metric operators
• Tridiagonality, supersymmetry and non self-adjoint Hamiltonians
• Solvable dilation model of $\mathcal{P}\mathcal{T}$-symmetric systems
• Experimental investigation of uncertainty relations for non-Hermitian operators
• Integrable nonlinear Klein-Gordon systems with $\mathcal{P}\mathcal{T}$ nonlocality and/or space-time exchange nonlocality
• Bosonic pair creation and the Schiff-Snyder-Weinberg effect
• On regular and chaotic dynamics of a non-$\mathcal{P}\mathcal{T}$-symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain
• Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians
• Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics
• N-site-lattice analogues of $V(x)=i{x}^3$
• The Pauli equation with complex boundary conditions
• A concise review on pseudo-bosons, pseudo-fermions and their relatives
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Critical coupling and coherent perfect absorption for ranges of energies due to a complex gain and loss symmetric system
• Generalized gauge transformation with $PT$-symmetric non-unitary operator and classical correspondence of non-Hermitian Hamiltonian for a periodically driven system
• Random-matrix theory of amplifying and absorbing resonators with PT or PTT' symmetry
• Appearances of pseudo-bosons from Black-Scholes equation
• Semi-regular biorthogonal pairs and generalized Riesz bases
• Integrable Minisuperspace Models with Liouville Field: Energy Density Self-Adjointness and Semiclassical Wave Packets
• Integrable coupled Li$\acute{e}$nard-type systems with balanced loss and gain
• Time-dependent C-operators as Lewis-Riesenfeld invariants in non-Hermitian theories
• Equivalence and superposition of real and imaginary quasiperiodicities
• Pseudo-Hermitian coherent states under the generalized quantum condition with position-dependent mass
• On elements of the Lax-Phillips scattering scheme for PT-symmetric operators
• Emergent non-Hermitian localization phenomena in the synthetic space of zero-dimensional bosonic systems
• WKB Analysis of PT-Symmetric Sturm-Liouville problems. II
• On the spectrum of DW Hamiltonian of quantum SU(2) gauge field
• Damping and Pseudo-fermions
• Complex BPS Skyrmions with real energy
• A generalized family of discrete PT-symmetric square wells
• Non-Hermitian Yukawa interactions of fermions with axions: potential microscopic origin and dynamical mass generation
• Anomalous mechanisms of the loss of observability in non-Hermitian quantum models
• Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations
• Quantum Metrology with Higher-order Exceptional Points in Atom-cavity Magnonics
• Schwinger-Dyson equations and mass generation for an axion theory with a PT symmetric Yukawa fermion interaction
• Modal Purcell factor in $\mathcal{P}\mathcal{T}$-symmetric waveguides
• WKB Analysis of PT-Symmetric Sturm-Liouville problems
• Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials
• Deconstructing non-Dirac-hermitian supersymmetric quantum systems
• Pseudo entropy and pseudo-Hermiticity in quantum field theories
• Broadband and Wide-Angle Invisibility with PT-Symmetric 2D-Weyl Semimetal
• Scattering states of a particle, with position-dependent mass, in a $\mathcal{P}\mathcal{T}$ symmetric heterojunction
• Transmission across non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric quantum dots and ladders
• The cryptohermitian smeared-coordinate representation of wave functions
• PT-symmetrically deformed shock waves
• PT-symmetric quantum systems with positive P
• Comparing Hermitian and Non-Hermitian Quantum Electrodynamics
• Weak Measurements in Non-Hermitian Systems
• Topology-controlled spectra of imaginary cubic oscillators in the large-L approach
• Finite-dimensional pseudo-bosons: a non-Hermitian version of the truncated harmonic oscillator
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• Sum rule for the pseudo-Rényi entropy
• Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction : General results and a case study with Landau Hamiltonian
• Parity-time symmetric systems with memory
• Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM
• Perturbative approach for strong and weakly coupled time-dependent non-Hermitian quantum systems
• Anyonic $\mathcal{P}\mathcal{T}$ symmetry, drifting potentials and non-Hermitian delocalization
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Real spectra in one-dimensional single-band non-Hermitian Hamiltonians
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts
• Systematics of quasi-Hermitian representations of non-Hermitian quantum models
• The coupled-cluster approach to quantum many-body problem in a three-Hilbert-space reinterpretation
• Quantum, noncommutative and MOND corrections to the entropic law of gravitation
• Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians
• Paths of unitary access to exceptional points
• PT-symmetric gain and loss in a rotating Bose-Einstein condensate
• Coexistence of non-Hermitian skin effect and extended states in one-dimensional nonreciprocal lattices
• Nonuniqueness of the C operator in PT-symmetric quantum mechanics
• Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation
• PT-Symmetry Breaking Transitions in Polymeric Systems
• Non self-adjoint Hamiltonians with complex eigenvalues
• Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
• General PT-Symmetric Matrices
• Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
• Internal nonlocality in generally dilated Hermiticity
• Non-Hermitian Model for Asymmetrical Tunneling
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions
• Topological invariants of complex energy plane in non-Hermitian systems
• A new non-Hermitian E2-quasi-exactly solvable model
• Rigged Hilbert Space Approach for Non-Hermite Systems with Positive Definite Metric
• Linear Response for pseudo-Hermitian Hamiltonian Systems: Application to PT-Symmetric Qubits
• Abstract ladder operators and their applications
• Galois conjugates of String-net Model
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Quantum-optical implementation of non-Hermitian potentials for asymmetric scattering
• Nonlinear transition between PT-symmetric and PT-broken modes in coupled fiber lasers
• Transition Probabilities for Flavor Eigenstates of Non-Hermitian Hamiltonians in the PT-Broken Phase
• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
• Enhanced frequency and temperature estimation by a $\mathcal{P}\mathcal{T}$-symmetric quantum oscillator
• Nonlinear supersymmetry as a hidden symmetry
• Broken Hermiticity phase transition in Bose-Hubbard model
• Quantum mechanical settings inspired by RLC circuits
• Topological phase transition in fluctuating imaginary gauge fields
• An Algebraic PT-Symmetric Quantum Theory with a Maximal Mass
• Exceptional points and quantum correlations in precise measurements
• Time invariant $\mathcal{P}\mathcal{T}$-product and phase locking in $\mathcal{P}\mathcal{T}$-symmetric lattice models
• Theory Including Future Not Excluded -- Formulation of Complex Action Theory II --
• On S-matrix of Schrodinger Operators with Non-Symmetric Zero-Range Potentials
• Quantum star-graph analogues of PT-symmetric square wells. II: Spectra
• Gibbs states defined by biorthogonal sequences
• PT-symmetric model with an interplay between kinematical and dynamical non-localities
• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
• Non self-adjoint operators with real spectra and extensions of quantum mechanics
• Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry
• Hermitian - non-Hermitian interfaces in quantum theory
• Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime
• Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction
• New families of non-parity-time-symmetric complex potentials with all-real spectra
• Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections
• Exceptional points and pseudo-Hermiticity in real potential scattering
• Hermitian--to--quasi-Hermitian quantum phase transitions
• Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
• Quantization of Big Bang in crypto-Hermitian Heisenberg picture
• Time reversal of a discrete system coupled to a continuum based on non-Hermitian flip
• Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials
• Integral Transforms and $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians
• Three perspectives on entropy dynamics in a non-Hermitian two-state system
• Adiabatic Series Expansion and Higher-Order Semiclassical Approximations in Scattering Theory
• Unitarity through PT symmetry in Quantum Quadratic Gravity
• An introductory review on resource theories of generalized nonclassical light
• Essential implications of similarities in non-Hermitian systems
• Pseudo-Hermitian extensions of the harmonic and isotonic oscillators
• Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions
• Resonant-amplified and invisible Bragg scattering based on spin coalescing modes
• Susy for non-Hermitian Hamiltonians, with a view to coherent states
• CPT-conserved effective mass Hamiltonians through first and higher order charge operator C in a supersymmetric framework
• A generalized Swanson Hamiltonian in a second-derivative pseudo-supersymmetric framework
• On-chip optical isolator and nonreciprocal parity-time symmetry induced by stimulated Brillouin scattering
• Classical-quantum correspondence for two-level pseudo-Hermitian systems
• Achieving the Quantum Fisher Information Bound in Pseudo-Hermitian Sensors
• PT-symmetry breaking and maximal chirality in a nonuniform PT-symmetric ring
• Manifestation of Superposition and Coherence in $\mathcal{P}\mathcal{T}$-symmetry through the $\eta$-inner Product
• New version of pseudo-hermiticity in the two-sided deformation of Heisenberg algebra
• Complex tridiagonal quantum Hamiltonians and matrix continued fractions
• Krein signature for instability of $\mathcal{P}\mathcal{T}$-symmetric states
• Multicomplex solitons
• Cumulants of time-integrated observables of closed quantum systems and PT-symmetry, with an application to the quantum Ising chain
• Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory
• Stability of time-periodic $\mathcal{P}\mathcal{T}$ and anti-$\mathcal{P}\mathcal{T}$-symmetric Hamiltonians with different periodicities
• Schrodinger Operators with Non-Symmetric Zero-Range Potentials
• Arnold's potentials and quantum catastrophes II
• Scattering in a varying mass PT symmetric double heterojunction
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Initial states of qubit-boson models leading to conserved quantities
• Artificial Hawking radiation, weak pseudo-Hermiticity and Weyl semimetal blackhole analogy
• Quantum control and the challenge of non-Hermitian model-building
• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space
• Pseudo-Hermitian and PT -symmetric quantum systems with energy-dependent potentials: Bound-state solutions and energy spectra
• Topological state permutations in time-modulated non-Hermitian multiqubit systems with suppressed non-adiabatic transitions
• Dissolution of the non-Hermitian skin effect in one-dimensional lattices with linearly varying nonreciprocal hopping
• On the Pauli group on 2-qubits in dynamical systems with pseudofermions
• Robustness of entanglement in a non-Hermitian cavity-optomechanical system even away from exceptional points
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation
• Equivalence Principle for Quantum Mechanics in the Heisenberg Picture
• Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables
• Schrieffer-Wolff transformation for non-Hermitian systems: application for $\mathcal{P}\mathcal{T}$-symmetric circuit QED
• Conserved quantities in non-Hermitian systems via vectorization method
• Quantum singularities in a solvable toy model
• Emerging (2+1)D electrodynamics and topological instanton in pseudo-Hermitian two-level systems
• Nonlinear n-Pseudo Fermions
• Quantum Mechanics in Curved Space(time) with a Noncommutative Geometric Perspective
• Markov constant and quantum instabilities
• Non-Hermitian coherent states for finite-dimensional systems
• Bi-coherent states as generalized eigenstates of the position and the momentum operators
• Raising the $\mathcal{P}\mathcal{T}$ transition threshold by strong coupling to neutral chains
• Hidden symmetries in non-self-adjoint graphs
• Pseudo-Hermitian systems with PT-symmetry: Degeneracy and Krein space
• Crypto-Hermitian Approach to the Klein-Gordon Equation
• Quantization of Second Order Fermions
• Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation
• Symmetries and invariants for non-Hermitian Hamiltonians
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Quantum geometric tensors from sub-bundle geometry
• Uncertainty relations: curiosities and inconsistencies
• Quantum phase transitions mediated by clustered non-Hermitian degeneracies
• Two dimensional non-Hermitian harmonic oscillator: coherent states
• Dual opposing quadrature-PT symmetry
• Reciprocity in parity violating non-Hermitian systems
• Remarks on the preservation of no-signaling principle in parity-time-symmetric quantum mechanics
• The large-g observability of the low-lying energies in the strongly singular potentials $V(x)=x^2+g^2/x^6$ after their PT-symmetric regularization
• Extracting the internal nonlocality from the dilated Hermiticity
• Witnessing criticality in non-Hermitian systems via entropic uncertainty relation
• Simulation of exceptional-point systems on quantum computers for quantum sensing
• Intrinsic exceptional point -- a challenge in quantum theory
• Ad hoc physical Hilbert spaces in Quantum Mechanics
• Moduli spaces for PT-regularized solitons
• Mathematical and physical meaning of the crossings of energy levels in $\mathcal{P}\mathcal{T}-$symmetric systems
• Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra
• Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models
• Perturbative expansion of irreversible works in symmetric and asymmetric processes
• Lax-Phillips scattering theory for PT-symmetric ρ-perturbed operators
• Effects of superradiance on relativistic Foldy-Wouthuysen densities
• Quantum implementation of non-unitary operations with biorthogonal representations
• Features, paradoxes and amendments of perturbative non-Hermitian quantum mechanics
• Topological spin textures in electronic non-Hermitian systems
• Few-grid-point simulations of Big Bang singularity in quantum cosmology
• Multi-well log-anharmonic oscillators
• Interference of non-Hermiticity with Hermiticity at exceptional points
• Realization of Lie algebras of high dimension via pseudo-bosonic operators
• Eigenvalues of non-hermitian matrices: a dynamical and an iterative approach. Application to a truncated Swanson model
• Construction of maximally non-Hermitian potentials under unbroken PT-symmetry constraint
• Planarizable Supersymmetric Quantum Toboggans
• Canonical quantization for Equilibrium Thermodynamics
• On a class of $J$-self-adjoint operators with empty resolvent set
• Phase transitions in quasi-Hermitian quantum models at exceptional points of order four
• Second-order pseudo-Hermitian spin-$1/2$ bosons
• Wigner multiplets in QFT: from Wigner degeneracy to Elko fields
• Exploring Metamaterial Lasers through Non-Hermitian Scattering Formalism
• Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models
• Edge states and persistent current in a PT-symmetric extended Su-Schrieffer-Heeger model with generic boundary conditions
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
• Pseudo-Hermitian systems, involutive symmetries and pseudofermions
• Nonlocality of Observables in Quasi-Hermitian Quantum Theory
• Quantum backreaction effect in optical solitons
• Topological invariant of velocity field in quantum systems
• Classical Hamiltonian Systems with Balanced loss and gain
• Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities
• Some results on the rotated infinitely deep potential and its coherent states
• Decomposition of a system in pseudo-Hermitian quantum mechanics
• Consistent treatment of quantum systems with a time-dependent Hilbert space
• Non-self-adjoint resolutions of the identity and associated operators
• Complementarity versus coordinate transformations: mapping between pseudo-Hermiticity and weak pseudo-Hermiticity
• Momentum relation and classical limit in the future-not-included complex action theory
• Nonlinear stationary states in PT-symmetric lattices
• A Hermitian bypass to the non-Hermitian quantum theory
• Supersymmetric $\eta$ operators
• PT-symmetry enabled stable modes in multi-core fiber
• Gaussian eigenstate pinning in non-Hermitian quantum mechanics
• Path-integral Bosonization of $d=2$ PT symmetric models
• Real spectra and phase transition of skin effect in nonreciprocal systems
• Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^2-g{x}^4+a/x^2$
• Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators
• Pseudo-bosons and bi-coherent states out of $\Lc^2(\mathbb{R})$
• Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
• On the finite-zone periodic PT-symmetric potentials
• Bosonic Spin-1 SOPHY
• Analytical construction of soliton families in one- and two-dimensional nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials
• Quantum mechanics using two auxiliary inner products
• Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics
• A weak pseudo-Hermitian two band model, artificial Hawking radiation and tunneling
• A Hamiltonian-like formalism that treats one spatial coordinate -- rather than time -- differently
• Swanson Hamiltonian: non-PT-symmetry phase
• Infinite series of time-dependent Dyson maps
• To quantify the difference of $\eta$-inner products in $\mathcal{P}\mathcal{T}$-symmetric theory
• Instability of an integrable nonlocal NLS
• A unifying E2-quasi-exactly solvable model
• Coulomb potential and the paradoxes of PT-symmetrization
• Quantum phase transitions and entanglement entropy in a non-Hermitian Jaynes-Cummings model
• Pseudo-bosons for the $D_2$ type quantum Calogero model
• Functional renormalization group for non-Hermitian and $\mathcal{P}\mathcal{T}$-symmetric systems
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory
• Generalised Geometric Phase
• Balancing gain and loss in symmetrised multi-well potentials
• Regular biorthogonal pairs and Psuedo-bosonic operators
• Reply to the Comment on `The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
• Reduced dynamics in quasi-Hermitian systems
• A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases
• Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians
• Evolution operator for time-dependent non-Hermitian Hamiltonians
• Algebraic construction of associated functions of nondiagonalizable models with anharmonic oscillator complex interaction
• Non-Hermitian Gravitational Wave Scattering
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• The geometry properties of parity and time reversal operators in two dimensional spaces
• PT-symmetric quantum models living in an auxiliary Pontryagin space
• Geometric criterion of topological phase transition for non-Hermitian systems
• Symbolic-manipulation constructions of Hilbert-space metrics in quantum mechanics
• Foundational aspects of spinor structures and exotic spinors
• Physical constraints on effective non-Hermitian systems
• Mutual compatibility/incompatibility of quasi-Hermitian quantum observables
• Emergence and localization of exceptional points in an exactly solvable toy model
• Euclidean-to-Lorentzian wormhole transition and gravitational symmetry breaking in the Sachdev-Ye-Kitaev model
• Emergence of Hermitian topology from non-Hermitian knots
• Regularization of energy-dependent pointlike interactions in 1D quantum mechanics
• Floquet spin textures in optically pumped non-Hermitian surface states
• Non-Hermitian Bose-Hubbard-like quantum models
• Triple exceptional point with unitary paths of unfolding in a three-site fermionic Swanson-like model
• Exploring Spectral Singularities in Dirac Semimetals: The Role of Non-Hermitian Physics and Dichroism
• Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating
• Covariant formulation of the Berry connection in non-Hermitian systems
• Operator based approach to PT-symmetric problems on a wedge-shaped contour
• $\mathcal{P}\mathcal{T}-$Symmetric Two-Level Open Quantum Systems: Information Theoretic Facets
• Construction of non-PT-symmetric complex potentials with all-real spectra
• Pursuing complex optical energy through Exceptional Points of Degeneracy (EPDs)
• Quantization of pseudo-hermitian systems
• Hilbert space representation for quasi-Hermitian position-deformed Heisenberg algebra and Path integral formulation
• Anisotropy and asymptotic degeneracy of the physical-Hilbert-space inner-product metrics in an exactly solvable crypto-unitary quantum model
• The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions
• Some invariant biorthogonal sets with an application to coherent states