Compactons in PT \mathcal{P}\mathcal{T} -symmetric generalized Korteweg-de Vries equations

This paper considers the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric extensions of the equations examined by Cooper, Shepard and Sodano. From the scaling properties of the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric equations a general theorem relating the energy, momentum and velocity of any solitarywave solution of the generalized KdV equation is derived. We also discuss the stability of the compacton solution as a function of the parameters affecting the nonlinearities.     

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.     

Twist deformations of the supersymmetric quantum mechanics

The $ \mathcal{N} $ \mathcal{N} -extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even $ \mathcal{N} $ \mathcal{N} one can identify the 1D $ \mathcal{N} $ \mathcal{N} -extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main differences between an abelian twist defined in terms of fermionic operators and an abelian twist defined in terms of bosonic operators are discussed.     

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.     

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.     

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.     

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.     

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.     

PT\mathcal{PT}-Symmetry in (Generalized) Effect Algebras

We show that an η ₊-pseudo-Hermitian operator for some metric operator η ₊ of a quantum system described by a Hilbert space H{\mathcal{H}} yields an isomorphism between the partially ordered commutative group of linear maps on H{\mathcal{H}} and the partially ordered commutative group of linear maps on H_r+{\mathcal{H}}_{\rho_{+}}. The same applies to the generalized effect algebras of positive operators and to the effect algebras of c-bounded positive operators on the respective Hilbert spaces H{\mathcal{H}} and H_r+{\mathcal{H}}_{\rho_{+}}. Hence, from the standpoint of (generalized) effect algebra theory both representations of our quantum system coincide.     

Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators H ^(±) is chosen antilinear. Secondly, both these components of a super-Hamiltonian $ \mathcal{H} $ \mathcal{H} are defined along certain topologically non-trivial complex curves r ^(±)(x) which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map $ \mathcal{T} $ \mathcal{T} between ‘tobogganic’ partner curves r ⁽⁺⁾(x) and r ⁽⁻⁾(x) is emphasized.     

Explaining GRB in the light of energy deposition rate for v+ [`(n)] \bar \nu → e+ + e? in a compact star

We have studied the v+ $ \bar \nu $ \bar \nu → e ⁺ + e ⁻ energy deposition rate near a rotating compact star which is important for the study of gamma ray bursts (GRB). The General relativistic (GR) and rotational effects increase the efficiency of the process immensely. The rotational effect also brings about an asymmetry in the deposition rate of the star.     

The p/π ratio pT dependence in the RHIC range of baryo-chemical potential

The BRAHMS measurement of proton-to-pion ratios in Au+Au and p+p collisions at $ \sqrt {s_{NN} } $ \sqrt {s_{NN} } = 62.4 GeV and $ \sqrt {s_{NN} } $ \sqrt {s_{NN} } = 200 GeV is presented as a function of transverse momentum and collision centrality within the pseudorapidity range 0 ≤ η ≤ 3. The baryo-chemical potential, μ _B , for the indicated data spans from μ _B ≈ 26 MeV ($ \sqrt {s_{NN} } $ \sqrt {s_{NN} } 200 GeV, η ≈ 0) to μ _B ∼ 260 MeV ($ \sqrt {s_{NN} } $ \sqrt {s_{NN} } = 62.4 GeV, η ≈ 3) [1]. The p/πratio measured for Au+Au system at $ \sqrt {s_{NN} } $ \sqrt {s_{NN} } 62.4 GeV, η ≈ 3 reaches astounding value of 8–10 at p _T > 1.5 GeV/c. For these energy and pseudorapidity interval no centrality dependency of p/π ratio is observed. Moreover, the baryon-to-meson ratio of nucleus-nucleus data are consistent with results obtained for p+p interactions.     

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.     

Classical and quantum mechanics of complex hamiltonian systems: An extended complex phase space approach

Certain aspects of classical and quantum mechanics of complex Hamiltonian systems in one dimension investigated within the framework of an extended complex phase space approach, characterized by the transformation x = x ₁ + ip ₂, p = p ₁ + ix ₂, are revisited. It is argued that Carl Bender inducted $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry in the studies of complex power potentials as a particular case of the present general framework in which two additional degrees of freedom are produced by extending each coordinate and momentum into complex planes. With a view to account for the subjective component of physical reality inherent in the collected data, e.g., using a Chevreul (hand-held) pendulum, a generalization of the Hamilton’s principle of least action is suggested.     

Maximal possible accretion rates for slim disks

It was proved in the previous work that there must be a maximal possible accretion rate $ \dot M_{max} $ \dot M_{max} for a slim disk. Here we discuss how the value of $ \dot M_{max} $ \dot M_{max} depends on the two fundamental parameters of the disk, namely the mass of the central black hole M and the viscosity parameter α. It is shown that $ \dot M_{max} $ \dot M_{max} increases with decreasing α, but is almost independent of M if $ \dot M_{max} $ \dot M_{max} is measured by the Eddington accretion rate $ \dot M_{Edd} $ \dot M_{Edd} , which is in turn proportional to M.     

The model-independent analysis indications for spectroscopy of scalar mesons. Proof for the K0* (900)

In a model-independent approach the data on ππ → ππ, K $ \bar K $ \bar K , ηη, ηη′ in the I ^G J ^PC = 0⁺0⁺⁺ channel and on the Kπ scattering in the $ I\left( {J^P } \right) = \frac{1} {2}\left( {0^ + } \right) $ I\left( {J^P } \right) = \frac{1} {2}\left( {0^ + } \right) channel are analyzed jointly for studying the status and QCD nature of the f ₀- and the K*₀-mesons. It is shown that in the 1500-MeV region, there are two states, wide (interpreted as a glueball) and narrow (q $ \bar q $ \bar q ). In the Kπ-scattering data analysis, the proof for the K*₀(900) is given.     

The masses of charged leptons and quarks from superposition self-interference of their Dirac fields

Without Higgs field interaction, accurate pole mass values are obtained for the charged leptons and quarks from a Z₃-symmetric linear superposition self-interference of the Dirac fields in the effective free-field Lagrangian. The charged lepton and quark pole masses evidence the discrete Z₃ symmetry, the theoretical-experimental deviations δm/m are $ \mathcal{O} $ \mathcal{O} (10⁻⁵) for all three charged leptons, and the quark pole masses are in very satisfactory overall agreement with the experimental data.     

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.     

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.     

On nuclear states of [`(K)] \bar K mesons

The search for nuclear states of $ \bar K $ \bar K mesons poses interesting problems for the nuclear and low energy hadron physics: the behavior of tightly bound nuclear systems with strongly correlated impurities, the new kind of binding mechanisms and the extension of effective low energy theories to the strange sector. These problems are briefly presented and a method of variational calculation of the binding energies is discussed.