Numerical Evidence that the Perturbation Expansion for a Non-Hermitian $\mathcal{P}\mathcal{T}$-Symmetric Hamiltonian is Stieltjes

Abstract

Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{P}\mathcal{T}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=p^2+1/4x^2+i\lambda x^3$ recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh-Schrödinger perturbation series in powers of ${\lambda }^2$ for the ground-state energy were calculated. Padé-summation and Padé-prediction techniques recently described by Weniger are applied to this perturbation series. The qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator, which is known to be a Stieltjes series.

20 pages, 0 figures

References in corpus (11)

• A new PT symmetric complex Hamiltonian with a real spectra
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• Shape invariant potentials with PT symmetry
• On the relation between Stokes multipliers and the T-Q systems of conformal field theory
• Variational Ansatz for PT-Symmetric Quantum Mechanics
• The energy spectrum of complex periodic potentials of the Kronig-Penney type
• Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm
• Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients
• Asymptotic Improvement of Resummation and Perturbative Predictions in Quantum Field Theory
• Pade-Improved Estimates of Hadronic Higgs Decay Rates
• Pade-Improvement of CP-odd Higgs Decay Rate into Two Gluons

Cited by in corpus (39)

• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
• Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators
• Bound states of the Dirac equation for the PT-symmetric generalized Hulthen potential by the Nikiforov-Uvarov method
• Nonperturbative Quantum Physics from Low-Order Perturbation Theory
• Exponential Type Complex and non-Hermitian Potentials in PT-Symmetric Quantum Mechanics
• Imaginary Cubic Perturbation: Numerical and Analytic Study
• Localizability of Tachyonic Particles and Neutrinoless Double Beta Decay
• Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by Č\'ıžek, Zamastil, and Skála. I. Algebraic Theory
• Ordinary versus PT-symmetric ${\varphi }^3$ quantum field theory
• Bound States of Non-Hermitian Quantum Field Theories
• Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree
• $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum
• On the Analyticity of Laguerre Series
• From Generalized Dirac Equations to a Candidate for Dark Energy
• Solvable simulation of a double-well problem in PT symmetric quantum mechanics
• Two-Point Green's Function in PT-Symmetric Theories
• Bernstein spectral method for quasinormal modes and other eigenvalue problems
• Eigenvalues of PT-symmetric oscillators with polynomial potentials
• Pseudo-Hermitian Quantum Dynamics of Tachyonic Spin-1/2 Particles
• The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions
• Dynamics of finite dimensional non-hermitian systems with indefinite metric
• Structure, Time Propagation and Dissipative Terms for Resonances
• The spectrum of the cubic oscillator
• Order-dependent mappings: strong coupling behaviour from weak coupling expansions in non-Hermitian theories
• Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series
• Neutrino Helicity Reversal and Fundamental Symmetries
• Non-perturbative calculations for the effective potential of the $PT$ symmetric and non-Hermitian $(-g{\varphi }^4)$ field theoretic model
• On the Bessel solution of Kepler's equation
• New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory
• Dirac Hamiltonian with Imaginary Mass and Induced Helicity-Dependence by Indefinite Metric
• Calculation of the Characteristic Functions of Anharmonic Oscillators
• Bender-Wu singularities
• Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213)
• Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration
• The Spherical Tensor Gradient Operator
• Neutrino Pair Cerenkov Radiation for Tachyonic Neutrinos
• Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator
• Convergence Analysis of the Summation of the Euler Series by Padé Approximants and the Delta Transformation