September 2012
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## Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

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)

## Complex Covariance

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)

## PT-Symmetric Talbot Effects

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)

## PT-Symmetric Electronics

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)

## Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator

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)

## Bragg solitons in nonlinear PT-symmetric periodic potentials

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)

## Non-hermitian model for resonant cavities coupled by a chiral mirror

Roberto Baginski B. Santos

Inspired by a recently observed asymmetry in the transmission of circularly polarized light through a metamaterial, we present a non-hermitian PT-symmetric quantum model to describe the interaction of the light fields in two resonant cavities coupled via a 2D-chiral mirror. We compute the time evolution of the light fields in this model, find two sets of operators compatible with the hamiltonian in a delocalized representation, discover the energies of the system and show that the transmission probability predicted by the model is indeed asymmetric.

http://arxiv.org/abs/1209.0819
Quantum Physics (quant-ph)

## Spectral singularity and non-Hermitian PT-symmetric extension of $$A_{N-1}$$ type Calogero model without confining potential

We consider non-Hermitian PT-symmetric deformation of $$A_{N-1}$$ type Calogero model without confining potential to investigate the possible existence of spectral singularity. By considering the Wronskian between asymptotic incoming and outgoing scattering state wave functions, we found that there exist no spectral singularity in this model. We further explicitly show that the transmission coefficient vanishes and the reflection coefficient becomes unity for all values of the energy in such a momentum dependent non-Hermitian PT-symmetric model.