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## Invisibility in PT-symmetric complex crystals

Stefano Longhi

Bragg scattering in sinusoidal PT-symmetric complex crystals of finite thickness is theoretically investigated by the derivation of exact analytical expressions for reflection and transmission coefficients in terms of modified Bessel functions of first kind. The analytical results indicate that unidirectional invisibility, recently predicted for such crystals by coupled-mode theory [Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)], breaks down for crystals containing a large number of unit cells. In particular, for a given modulation depth in a shallow sinusoidal potential, three regimes are encountered as the crystal thickness is increased. At short lengths the crystal is reflectionless and invisible when probed from one side (unidirectional invisibility), whereas at intermediate lengths the crystal remains reflectionless but not invisible; for longer crystals both unidirectional reflectionless and invisibility properties are broken.

http://arxiv.org/abs/1111.3448
Quantum Physics (quant-ph)

## N-site-lattice analogues of $$V(x)=i x^3$$

Miloslav Znojil

Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain $${\cal D}$$ of parameters $$a$$ and $$z$$ of the model, the spectrum of energies is shown real (i.e., potentially, observable) and the unitarity of the evolution is shown mediated by the construction of a (non-unique) physical, ad hoc Hilbert space endowed with a nontrivial, Hamiltonian-dependent inner-product metric $$\Theta$$. Beyond $${\cal D}$$ the complex-energy curves are shown to form a “Fibonacci-numbered” geometric pattern and/or a “topologically complete” set of spectral loci. The dynamics-determining construction of the set of the eligible metrics is shown tractable by a combination of the computer-assisted algebra with the perturbation and extrapolation techniques. Confirming the expectation that for the local potentials the effect of the metric cannot be short-ranged.

http://arxiv.org/abs/1111.0484
Quantum Physics (quant-ph); Mathematical Physics (math-ph)