Singular Mapping for a $PT$-Symmetric Sinusoidal Optical Lattice at the Symmetry-Breaking Threshold

Abstract

A popular $PT$-symmetric optical potential (variation of the refractive index) that supports a variety of interesting and unusual phenomena is the imaginary exponential, the limiting case of the potential $V_0[\cos (2\pi x/a)+i\lambda \sin (2\pi x/a)]$ as $\lambda \to 1$, the symmetry-breaking point. For $\lambda <1$, when the spectrum is entirely real, there is a well-known mapping by a similarity transformation to an equivalent Hermitian potential. However, as $\lambda \to 1$, the spectrum, while remaining real, contains Jordan blocks in which eigenvalues and the corresponding eigenfunctions coincide. In this limit the similarity transformation becomes singular. Nonetheless, we show that the mapping from the original potential to its Hermitian counterpart can still be implemented; however, the inverse mapping breaks down. We also illuminate the role of Jordan associated functions in the original problem, showing that they map onto eigenfunctions in the associated Hermitian problem.

References in corpus (12)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Unidirectional Invisibility induced by PT-Symmetric Periodic Structures
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Invisibility in PT-symmetric complex crystals
• Spectral singularities and Bragg scattering in complex crystals
• PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold
• On the metric operator for the imaginary cubic oscillator
• Analytic Results for a PT-symmetric Optical Structure
• A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x
• Use of Equivalent Hermitian Hamiltonian for $PT$-Symmetric Sinusoidal Optical Lattices
• Unbounded C-symmetries and their nonuniqueness