Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics

Abstract

In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian $H=\frac{1}{2}{p}^2+\frac{1}{2}{x}^2+i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H=\frac{1}{2}{p}^2+\frac{1}{2}{x}^2-\epsilon x^4$ using nonperturbative methods.

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Cited by in corpus (55)

• Making Sense of Non-Hermitian Hamiltonians
• Complex Extension of Quantum Mechanics
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• Must a Hamiltonian be Hermitian?
• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Perturbation theory in a framework of iteration methods
• Scalar Quantum Field Theory with Cubic Interaction
• Dual PT-Symmetric Quantum Field Theories
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• Time evolution of non-Hermitian Hamiltonian systems
• PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Ghost Busting: PT-Symmetric Interpretation of the Lee Model
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• The C Operator in PT-Symmetric Quantum Theories
• Fermionic coherent states for pseudo-Hermitian two-level systems
• Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics
• Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well
• The C Operator in PT-Symmetric Quantum Field Theory Transforms as a Lorentz Scalar
• Giving up the ghost
• Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
• PT-Symmetric Quantum Mechanics: A Precise and Consistent Formulation
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• Nonunique C operator in PT Quantum Mechanics
• Probability Density in the Complex Plane
• PT-Symmetric Quantum Theory Defined in a Krein Space
• Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework
• Construction of PT-asymmetric non-Hermitian Hamiltonians with CPT-symmetry
• An alternative construction of the positive inner product for pseudo-Hermitian Hamiltonians: Examples
• Remarks on PT-pseudo-norm in PT-symmetric quantum mechanics
• Overcritical PT-symmetric square well potential in the Dirac equation
• Thermodynamics of Pseudo-Hermitian Systems in Equilibrium
• Unbounded C-symmetries and their nonuniqueness
• Linear representation of energy-dependent Hamiltonians
• Gauging non-Hermitian Hamiltonians
• Identification of observables in quantum toboggans
• A Perturbative Treatment Of A Generalized PT-Symmetric Quartic Anharmonic Oscillator
• N-site-lattice analogues of $V(x)=i{x}^3$
• On elements of the Lax-Phillips scattering scheme for PT-symmetric operators
• $\mathcal{P}\mathcal{T}$ symmetric models with nonlinear pseudo supersymmetry
• Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$
• Semiclassical analysis of a complex quartic Hamiltonian
• Nonuniqueness of the C operator in PT-symmetric quantum mechanics
• Pseudo-Hermiticity, PT-symmetry, and the Metric Operator
• Treating Ostrogradski instability for Gallilean invariant Chern Simon's model via \mathcal{PT} symmetry
• Comment on ``New ansatz for metric operator calculation in pseudo-Hermitian field theory''
• Unidirectional segregation of Bright-Bright soliton through a $\mathcal{P}\mathcal{T}$-symmetric potential
• Removal of instabilities of the higher derivative theories in the light of antilinearity
• Two dimensional non-Hermitian harmonic oscillator: coherent states
• Uncertainty relations: curiosities and inconsistencies
• Computation and Verification of Spectra for Non-Hermitian Systems
• On the simpleness of zeros of Stokes multipliers
• More on Exact $\CP$-Symmetric Quantum Mechanics
• Effects of complex parameters on classical trajectories of Hamiltonian systems