• Volume 73, Issue 2

August 2009,   pages  215-1

• Preface

• From $\mathcal{PT}$ -symmetric quantum mechanics to conformal field theory

One of the simplest examples of a $\mathcal{PT}$-symmetric quantum system is the scaling Yang–Lee model, a quantum field theory with cubic interaction and purely imaginary coupling. We give a historical review of some facts about this model in $d \leq 2$ dimensions, from its original definition in connection with phase transitions in the Ising model and its relevance to polymer physics, to the role it has played in studies of integrable quantum field theory and $\mathcal{PT}$-symmetric quantum mechanics. We also discuss some more general results on $\mathcal{PT}$-symmetric quantum mechanics and the ODE/IM correspondence, and mention applications to magnetic systems and cold atom physics.

• Reality and non-reality of the spectrum of $\mathcal{PT}$-symmetric operators: Operator-theoretic criteria

We generalize some recently established criteria for the reality and non-reality of the spectrum of some classes of $\mathcal{PT}$-symmetric Schrödinger operators. The criteria include cases of discrete spectra and continuous ones.

• 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 $\mathcal{PT}$ -) 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.

• Conduction bands in classical periodic potentials

The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time to perform the measurement. This paper considers the possibility that $\Delta E$, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behaviour of a classical particle in a one-dimensional periodic potential $V(x) = − \cos(x)$ is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviours: (i) The particle may hop from classically allowed site to nearest-neighbour classically allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunnelling processes or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunnelling. The classical conduction bands for this potential are determined numerically with high precision.

• 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 $\mathcal{PT}$-symmetric Hamiltonians in modelling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.

• The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry

The non-equivalence of the presence of antilinear symmetry and pseudo-Hermiticity is shown for bounded operators. Two appropriate examples are operators with non-empty residual spectrum. The class of operators for which the equivalence holds is extended to the spectral operators of scalar type. The importance of 𝐽-self-adjointness is stressed and new proofs using this property are provided.

• 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 $\mathcal{PT}$ 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.

• Sturm–Schrödinger equations: Formula for metric

Sturm–Schrödinger equations $H\psi = EW\psi$ with $H \neq H^{\dagger}$ and $W \neq W^{\dagger} \neq I$ are considered, with a weak point of the theory lying in the purely numerical matrix- inversion form of the double-series definition of the necessary metric operator 𝛩 in the physical Hilbert space of states [M Znojil, J. Phys. A: Math. Theor. 41, 215304 (2008)]. This shortcoming is removed here via an amended, single-series definition of 𝛩.

• Spectra of $\mathcal{PT}$-symmetric Hamiltonians on tobogganic contours

A non-standard generalization of the Bender potentials $x^{2} (ix^{\varepsilon})$ is suggested. The spectra are obtained numerically and some of their particular properties are discussed.

• Level crossings in complex two-dimensional potentials

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

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

We find that a non-differentiability occurring whether in real or imaginary part of a complex $\mathcal{PT}$-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 $\mathcal{PT}$ -symmetric potentials $ix|x|$ and $ix$, respectively.

• Spontaneous breakdown of $\mathcal{PT}$ symmetry in the complex Coulomb potential

The $\mathcal{PT}$ symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $\mathcal{PT}$ symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $\mathcal{PT}$ -symmetric if the 𝐿 parameter that corresponds to angular momentum in the Hermitian case is real. $\mathcal{PT}$ symmetry is spontaneously broken, however, for complex 𝐿 values of the form $L = − \dfrac{1}{2} + i\lambda$. In this case the potential remains $\mathcal{PT}$ -symmetric, while the two independent solutions are transformed to each other by the $\mathcal{PT}$ operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.

• Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of $\mathcal{PT}$ 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. $\mathcal{PT}$ symmetry may facilitate reconciling our approach to the requirement that the rationally extended potentials be singularity free. Some examples are shown.

• Solution of an analogous Schrödinger equation for $\mathcal{PT}$-symmetric sextic potential in two dimensions

We investigate the quasi-exact solutions of an analogous Schrödinger wave equation for two-dimensional non-Hermitian complex Hamiltonian systems within the framework of an extended complex phase space characterized by $x = x_{1} + ip_{3}$, $y = x_{2} + ip_{4}$, $p_{x} = p_{1} + ix_{3}$, $p_{y} = p_{2} + ix_{4}$. Explicit expressions for the energy eigenvalues and eigenfunctions for ground and first excited states of a two-dimensional $\mathcal{PT}$-symmetric sextic potential and some of its variants are obtained. The eigenvalue spectra are found to be real within some parametric domains.

• Particles versus fields in $\mathcal{PT}$-symmetrically deformed integrable systems

We review some recent results on how $\mathcal{PT}$ symmetry, that is a simultaneous time-reversal and parity transformation, can be used to construct new integrable models. Some complex valued multi-particle systems, such as deformations of the Calogero–Moser– Sutherland models, are shown to arise naturally from real valued field equations of non-linear integrable systems. Deformations of complex non-linear integrable field equations, some of them even allowing for compacton solutions, are also investigated. The integrabilty of various systems is established by means of the Painlevé test.

• Compactons in $\mathcal{PT}$-symmetric generalized Korteweg–de Vries equations

This paper considers the $\mathcal{PT}$-symmetric extensions of the equations examined by Cooper, Shepard and Sodano. From the scaling properties of the $\mathcal{PT}$-symmetric equations a general theorem relating the energy, momentum and velocity of any solitary-wave 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.

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

We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $\mathcal{PT}$ -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 $\mathcal{PT}$ - invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender–Dunne polynomials.

• 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 $H^{ (\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian $\mathcal{H}$ are defined along certain topologically non-trivial complex curves $r^{ (\pm)} (x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map $\mathcal{T}$ between `tobogganic' partner curves $r^{ (+)} (x)$ and r^{ (−)} (x)\$ is emphasized.

• Use of supersymmetric isospectral formalism to realistic quantum many-body problems

We propose a novel mathematical approach for the calculation of resonances in weakly bound systems. For any potential, families of strictly isospectral potentials (with very different shape) having desirable and adjustable features can be generated. For systems having no bound ground state, an isospectral potential with a bound state in the continuum is possible. The quasi-bound state in the original shallow potential will be effectively trapped in the deep well of the isospectral family, facilitating more accurate calculation of resonance energy. Application to 6He, 6Li and 6Be yield excellent results. Another application is the calculation of Efimov states in weakly bound three-body system. We present the result of 4He trimer, where the first excited state is claimed to be an Efimov state.

• List of participants

• Organizing Committee

• # Pramana – Journal of Physics

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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019