January 2014
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Day January 20, 2014

Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra

Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schroedinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy eigenspectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility

Ali Mostafazadeh

The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic approximation to H(t) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(t) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)