September 2014
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Day September 12, 2014

Interplay of disorder and PT-symmetry

C. Mejía-Cortés, M. I. Molina

We examine a one-dimensional PT-symmetric binary lattice in the presence of diagonal disorder. We focus on the wave transport phenomena of localized and extended input beams for this disordered system. In the pure PT-symmetric case, we derive an exact expression for the evolution of light localization in terms of the typical parameters of the system. In this case localization is enhanced as the gain and loss parameter in increased. In the presence of disorder, we observe that the presence of gain and loss inhibits (favors) the transport for localized (extended) excitations.
Optics (physics.optics)

Metric operators, generalized hermiticity and lattices of Hilbert spaces

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. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.
Mathematical Physics (math-ph)