July 2013
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Day July 24, 2013

PT-Symmetric Dimer of Coupled Nonlinear Oscillators

J. Cuevas, P.G. Kevrekidis, A. Saxena, A. Khare

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\”odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

http://arxiv.org/abs/1307.6047

Pattern Formation and Solitons (nlin.PS)

Some remarks on quasi-Hermitian operators

Jean-Pierre Antoine, Camillo Trapani

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

http://arxiv.org/abs/1307.5644
Mathematical Physics (math-ph)