Spontaneous breakdown of $$ \mathcal{P}\mathcal{T} $$ symmetry in the complex Coulomb potential

The $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric if the L parameter that corresponds to angular momentum in the Hermitian case is real. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry is spontaneously broken, however, for complex L values of the form L = −1/2 + iλ. In this case the potential remains $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric, while the two independent solutions are transformed to each other by the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.     

Scarcity of real discrete eigenvalues in non-analytic complex $$ \mathcal{P}\mathcal{T} $$-symmetric potentials

We find that a non-differentiability occurring whether in real or imaginary part of a complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials x ² and |x| by imaginary $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potentials ix/it|x| and ix, respectively.     

PT\mathcal {P}\mathcal {T}-Symmetric Klein-Gordon Oscillator

Parity-time (PT)(\mathcal {P}\mathcal {T}) symmetric Klein-Gordon oscillator is presented using PT\mathcal {P}\mathcal {T}-symmetric minimal substitution. It is shown that wave equation is exactly solvable, and energy spectrum is the same as that of Hermitian Klein-Gordon oscillator presented by Bruce and Minning. Landau problem of PT\mathcal {P}\mathcal {T}-symmetric Klein-Gordon oscillator is discussed.     

New quasi-exactly solvable Hermitian as well as non-Hermitian $$ \mathcal{P}\mathcal{T} $$-invariant potentials

We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials.     

The property of maximal transcendentality in the N \mathcal{N} = 4 SYM

We show results for the universal anomalous dimension γ_uni(j) of Wilson twist-2 operators in the $ \mathcal{N} $ \mathcal{N} = 4 Supersymmetric Yang-Mills theory in the first three orders of perturbation theory. These expressions are obtained by extracting the most complicated contributions from the corresponding anomalous dimensions in QCD.     

Complexification of three potential models — II

A new kind of $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} and non-$ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric complex potentials are constructed from a group theoretical viewpoint of the sl(2,C) potential algebras. The real eigenvalues and the corresponding regular eigenfunctions are also obtained. The results are compared with the ones obtained before.     

Isospectrality of conventional and new extended potentials,second-order supersymmetry and role of $$ \mathcal{P}\mathcal{T} $$ symmetry

We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry may facilitate reconciling our approach to the requirement that the rationally extended potentials be singularity free. Some examples are shown.     

Non-Hermitian Hamiltonians with a real spectrum and their physical applications

We present an evaluation of some recent attempts to understand the role of pseudo-Hermitian and $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric Hamiltonians in modelling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Level crossings in complex two-dimensional potentials

Two-dimensional $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric quantum-mechanical systems with the complex cubic potential V ₁₂ = x ² + y ² + igxy ² and the complex Hénon-Heiles potential V _HH = x ² +y ² +ig(xy ² −x ³/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of symmetry and the non- orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.     

Krein-space formulation of \mathcal{P}\mathcal{T} symmetry, \mathcal{C}\mathcal{P}\mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the $\mathcal{C}\mathcal{P}\mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics.     

On a few new quantization recipes using \mathcal{P}\mathcal{T}-symmetry

A review of a few recent developments in our analysis and applications of the coupled-channel version of the formalism of -symmetric quantum mechanics is given.     

Statistical features of quantum evolution

It is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian of the quantum system for an arbitrary pseudo-unitary (and hence $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -) quantum evolution. The result generalizes the time-energy uncertainty principle for pseudo-unitary quantum evolutions. Further, employing random matrix theory developed for pseudo-Hermitian systems, time correlation functions are studied in the framework of linear response theory. The results given here provide a quantum brachistochrone problem where the system will evolve in a thermodynamic environment with spectral complexity that can be modelled by random matrix theory.     

P T $\mathcal {P}\mathcal {T}$ -symmetry in Compact Phase Space for a Linear Hamiltonian

International Journal of Theoretical Physics - We study the time evolution of a $\mathcal {P}\mathcal {T}$ -symmetric, non-Hermitian quantum system for which the associated phase space is compact....     

\mathcal{P}\mathcal{T}-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach

To lowest order of perturbation theory we show that an equivalence can be established between a -symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian h. An important feature of h is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical quantities.     

To Quantify the Difference of η-Inner Products in P T $\mathcal {P}\mathcal {T}$ -Symmetric Theory

International Journal of Theoretical Physics - In this paper, we consider a typical continuous two dimensional $\mathcal {P}\mathcal {T}$ -symmetric Hamiltonian and propose two different approaches...     

Impact of P T $\mathcal {P}\mathcal {T}$ -Symmetric Operation on Concurrence and the First-Order Coherence

International Journal of Theoretical Physics - In this paper, we lock the focus in effect of $\mathcal {P}\mathcal {T}$ -symmetric operation on the dynamics of concurrence and the first-order...     

Analytic wave functions and energies for two-dimensional \mathcal{P}\mathcal{T} -symmetric quartic potentials

Analytic wave functions and the corresponding energies for a class of the $ \mathcal{P}\mathcal{T} $ -symmetric two-dimensional quartic potentials are found. The general form of the solutions is discussed.