Bikashkali Midya, Barnana Roy
Using an appropriate change of variable, the Schroedinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type \(X_m\) exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric and hyperbolic Scarf potentials whose bound state solutions are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these `rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.
http://arxiv.org/abs/1210.0119
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Carl M. Bender, Stefan Boettcher
The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the paper by Bowen et al are incorrectly applied and wrongly interpreted.
http://arxiv.org/abs/1210.0426
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Tomaz Prosen
We outline two general classes of examples of PT-symmetric quantum Liouvillian dynamics of open many qubit systems, namely interacting hard-core bosons (or more general XYZ-type spin 1/2 systems) with, either (i) pure dephasing noise, or (ii) having solely single particle/spin injection/absorption incoherent processes.
http://arxiv.org/abs/1209.6588
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Paolo Amore, Francisco M Fernández
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender’s argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Rep. Prog. Phys. 70 (2007) 947).
http://arxiv.org/abs/1209.6357
Quantum Physics (quant-ph)
Patrick Dorey, Clare Dunning, Roberto Tateo
A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than two. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain second-order cases, and extended to the higher-order problems.
http://arxiv.org/abs/1209.4736
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)
F. Kleefeld
According to some generalized correspondence principle the classical limit of a non-Hermitian Quantum theory describing quantum degrees of freedom is expected to be well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity has been developed by Einstein merely for real-valued space-time and four-momentum we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, yet also to the non-Hermitian Klein-Gordon and Dirac equations which are to lay the foundations of a non-Hermitian quantum theory.
http://arxiv.org/abs/1209.3472
Mathematical Physics (math-ph)
Hamidreza Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, Tsampikos Kottos
We show that complex PT-symmetric photonic lattices can lead to a new class of self-imaging Talbot effects. For this to occur, we find that the input field pattern, has to respect specific periodicities which are dictated by the symmetries of the system. While at the spontaneous PT-symmetry breaking point, the image revivals occur at Talbot lengths governed by the characteristics of the passive lattice, at the exact phase it depends on the gain and loss parameter thus allowing one to control the imaging process.
http://arxiv.org/abs/1209.2349
Optics (physics.optics); Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)
J. Schindler, Z. Lin, J. M. Lee, Hamidreza Ramezani, F. M. Ellis, Tsampikos Kottos
We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time PT operator: typical normal modes, temporal evolution, and scattering processes. Utilizing a Liouvilian formulation, we can define an underlying PT-symmetric Hamiltonian, which provides important insight for understanding the behavior of the system. When the PT-dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the PT-symmetry of the scattering target. Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves. Having an exact theory, and due to its relative experimental simplicity, PT-symmetric electronics offers new insights into the properties of PT-symmetric systems which are at the forefront of the research in mathematical physics and related fields.
http://arxiv.org/abs/1209.2347
Other Condensed Matter (cond-mat.other); Optics (physics.optics); Quantum Physics (quant-ph)
V. Chithiika Ruby, M. Senthilvelan, M. Lakshmanan
We carry out an exact quantization of a PT symmetric (reversible) Lienard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schrodinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.
http://arxiv.org/abs/1209.1182
Quantum Physics (quant-ph)
Mohammad-Ali Miri, Alejandro B. Aceves, Tsampikos Kottos, Vassilios Kovanis, Demetrios N. Christodoulides
It is shown that slow Bragg soliton solutions are possible in nonlinear complex parity-time (PT) symmetric periodic structures. Analysis indicates that the PT-symmetric component of the periodic optical refractive index can modify the grating band structure and hence the effective coupling between the forward and backward waves. Starting from a classical modified massive Thirring model, solitary wave solutions are obtained in closed form. The basic properties of these slow solitary waves and their dependence on their respective PT-symmetric gain/loss profile are then explored via numerical simulations.
http://arxiv.org/abs/1209.0787
Optics (physics.optics); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Quantum Physics (quant-ph)