XXXIII Workshop on Geometric Methods in Physics 29.06-5.07.2014

Ali Mostafazadeh

Spectral Singularities, Unidirectional Invisibility, and Dynamical Formulation of One-Dimensional Scattering Theory

Complex scattering potentials have surprising properties that real scattering potentials lack. Among these are spectral singularities and unidirectional invisibility. In the first part of this talk, I will survey the recent development in the study of physical meaning and applications of spectral singularities. In particular, I show that they provide the basic mathematical framework for all lasing and antilasing systems. In the second part of the talk, I outline a dynamical theory of potential scattering in one-dimension and offer an inverse scattering scheme for the construction of scattering potentials with desirable properties at a prescribed wavelength. This is based on the curious observation that given a possibly complex scattering potential v(x) we can construct a two-level non-stationary and non-Hermitian Hamiltonian H(t) whose S-matrix coincides with the transfer matrix of v(x). We use this approach to develop a complete perturbative description of the phenomenon of unidirectionally invisibility, construct multi-mode unidirectionally invisible potentials with wavelength-dependent direction of invisibility, and show that the application of the adiabatic approximation for H(t) coincides with the semiclassical (WKB) treatment of the scattering problem. A remarkable outcome of the latter result is the identification of the geometric part of the phase of the adiabatically evolving states with the pre-exponential factor of the WKB wave functions.

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A. Mostafazadeh, J. Phys. A 47, 125301 (2014); arXiv: 1401.4315.

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