PT Symmetry

Bender, Carl M.; Dorey, Patrick; Dunning, Clare; Fring, Andreas; Hook, Daniel W.; Jones, H. F.; Kuzhel, Sergii; Lévai, G.; Tateo, Roberto

doi:10.1142/q0178

Cited by 166 publications

(137 citation statements)

References 53 publications

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“…We recall [1] that a nonnegative self-adjoint extension (7). For this reason, the definition of extremal extensions can be rewritten as follows: let G 0 be determined by (10) (i) the self-adjoint operator Q and the ONB {e n } are determined uniquely in (11);…”

Section: Ii2 Regular Biorthogonal Sequences and Dual Grsmentioning

confidence: 99%

“…Remark III.1. The decomposition (16) Each fundamental decomposition (16) The operator C is called a C-symmetry operator and this notion is widely used in PT -symmetric approach in Quantum Mechanics [11].…”

Section: Iii1 Elements Of the Krein Spaces Theorymentioning

confidence: 99%

“…respectively. The functions {φ n } and {ψ n } are determined by (11) with the operator of multiplication Q = 2p(x) in L 2 (R) which anticommutes with P. Proposition III.6…”

Section: Iv2 Eigenfunctions Of Perturbed Anharmonic Oscillatormentioning

confidence: 99%

“…is real. It was conjectured [10] that the reality of eigenvalues of H is a consequence of This gave rise to a consistent development of theory of PT -symmetric Hamiltonians in Quantum Physics, see [4,5,11] and references therein.…”

Section: Introductionmentioning

confidence: 98%

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Generalized Riesz systems and orthonormal sequences in Krein spaces

Багарелло

Kużel

2020

J. Phys. A: Math. Theor.

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Section: Ii2 Regular Biorthogonal Sequences and Dual Grsmentioning

confidence: 99%

Section: Iii1 Elements Of the Krein Spaces Theorymentioning

confidence: 99%

“…respectively. The functions {φ n } and {ψ n } are determined by (11) with the operator of multiplication Q = 2p(x) in L 2 (R) which anticommutes with P. Proposition III.6…”

Section: Iv2 Eigenfunctions Of Perturbed Anharmonic Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Generalized Riesz systems and orthonormal sequences in Krein spaces

Багарелло

Kużel

2020

J. Phys. A: Math. Theor.

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“…Recently, for the microscopic description of the dissipation process within a quantum mechanics, a new formalism, known as complex spectral analysis [36][37][38][39] and non-Hemitian quantum mechanics [18,19,40,41], have been independently developed. In the complex spectral analysis, the functional space for a quantum state is extended to the rigged Hilbert space where the dual functional space is equipped with a bi-complete and bi-orthonormal basis set [21,22], so that the time-evolution generator, Hamiltonian or Liouvillian, has complex eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal

Tanaka

Kanki

2020

Physics

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We theoretically study the dynamical Casimir effect (DCE), i.e., parametric amplification of a quantum vacuum, in an optomechanical cavity interacting with a photonic crystal, which is considered to be an ideal system to study the microscopic dissipation effect on the DCE. Starting from a total Hamiltonian including the photonic band system as well as the optomechanical cavity, we have derived an effective Floquet-Liouvillian by applying the Floquet method and Brillouin-Wigner-Feshbach projection method. The microscopic dissipation effect is rigorously taken into account in terms of the energy-dependent self-energy. The obtained effective Floquet-Liouvillian exhibits the two competing instabilities, i.e., parametric and resonance instabilities, which determine the stationary mode as a result of the balance between them in the dissipative DCE. Solving the complex eigenvalue problem of the Floquet-Liouvillian, we have determined the stationary mode with vanishing values of the imaginary parts of the eigenvalues. We find a new non-local multimode DCE represented by a multimode Bogoliubov transformation of the cavity mode and the photon band. We show the practical advantage for the observation of DCE in that we can largely reduce the pump frequency when the cavity system is embedded in a narrow band photonic crystal with a bandgap.

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Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

Hao

Liang

2022

Annalen der Physik

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This paper demonstrates that the parity‐time (PT$PT$)‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be PT$PT$ symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase γn$\gamma _{n}$ for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with PT$PT$ symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle normalΔθH$\Delta \theta _{\text{H}}$ and Berry phase satisfy precisely the quantum‐classical correspondence, γn=(n+1/2)normalΔθH$\gamma _{n}= (n+1/2)\Delta \theta _{\text{H}}$.

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PT Symmetry

Abstract: Contribution from H. F. Jones I'll be more enthusiastic about encouraging thinking outside the box when there's evidence of any thinking going on inside it.

Cited by 166 publications

References 53 publications

Generalized Riesz systems and orthonormal sequences in Krein spaces

Generalized Riesz systems and orthonormal sequences in Krein spaces

The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal

Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

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