Perfect transmission scattering as a PT-symmetric spectral problem

Abstract

We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schrödinger operators with complex Robin boundary conditions.

Quantum star-graph analogues of PT-symmetric square wells

“…In all of these "non-real-spectrum" cases it seems necessary to discard the underlying Hamiltonian H = H (q) (α) as leading to non-unitary evolution of the quantum system in question. At the same time, many of the formal (e.g., solvability) as well as phenomenological (e.g., scattering-related [21]) features of these and similar models might seem appealing enough. For this reason, let us now describe, briefly, one of the recently discovered and more or less universal remedies of the apparent complex-energy shortcoming.…”

Optical waveguide arrays: quantum effects and PT symmetry breaking

“…The region of parameter space where all energy eigenvalues of a PT -symmetric Hamiltonian are real is traditionally called the PT -symmetric region, and the emergence of complex conjugate eigenvalues that accompanies departure from this region is called PT -symmetry breaking. Since the effective potential in an optical waveguide array is given by the local (complex) index of refraction, properties of PT Hamiltonians have led to predictions of new optical phenomenon such as Bloch oscillations in complex crystals [34], perfect transmission [35] and a perfect absorber of coherent waves [36,37], PT -symmetric Dirac equation [38], induced quantum coherence between BoseEinstein condensates [39], and topologically protected midgap states in honeycomb lattices [40,41]. The stability of nonlinear solitions in PT -symmetric systems has also been investigated [42,43].…”

Lax-Phillips scattering theory for PT-symmetric ρ-perturbed operators

“…with PT -symmetric potential q(x), i.e., q(x) = q(−x) were studied by different methods [1,8,11,12,18,20,24,25]. In particular, scattering on the PT -symmetric Coulomb potential was studied on a trajectory of the complex plane [18]; discretization methods were used for getting the explicit formulae for the reflection and transmission coefficients [24,25]; the relationship between PT -symmetric Hamiltonians and reflectionless scattering systems was discussed [1,12]; spectral singularities were characterized in terms of reflection and transmission coefficients [20].…”

“…In particular, scattering on the PT -symmetric Coulomb potential was studied on a trajectory of the complex plane [18]; discretization methods were used for getting the explicit formulae for the reflection and transmission coefficients [24,25]; the relationship between PT -symmetric Hamiltonians and reflectionless scattering systems was discussed [1,12]; spectral singularities were characterized in terms of reflection and transmission coefficients [20]. If the potential q(x) in (1.1) is local, that is, if its support is contained in an interval (−ρ, ρ), then the corresponding traveling wave functions have the form: are the left and right transmission coefficients, respectively and k > 0.…”