One-dimensional crystal with a complex periodic potential

Abstract

A one-dimensional crystal model is constructed with a complex periodic potential. A wave function solution for the crystal model is derived without relying on Bloch functions. The new wave function solution of this model is shown to correspond to the solution for the probability amplitude of a two-level system. The energy discriminant is evaluated using an analytic formula derived from the probability amplitude solution, and based on an expansion parameter related to the energy and potential amplitude. From the wave function energy discriminant the crystal band structure is derived and related to standard energy bands and gaps. It is also shown that several of the properties of the two-level system apply to the one-dimensional crystal model. The two-level system solution which evolves in time is shown to manifest as a spatial configuration of the one-dimensional crystal model. The sensitivity of the wave function probability density is interpreted in the context of the new solution. The spatial configuration of the wave function, and the appearance of a long wavelength in the wave function probability density is explained in terms of the properties of Bessel functions.

$\mathcal{PT}$ -Symmetric Periodic Optical Potentials

“…We emphasize that these peculiar features are a direct consequence of the non-orthogonality of the PT Floquet-Bloch functions. In fact, the usual orthogonality condition (15) (that is valid in real crystals) is no longer observed in PT symmetric lattices. This skewness of the FB modes is an inherent characteristic of PT symmetric periodic potentials and has important consequences on their algebra.…”

“…However, since the spectral problem (8) is not selfadjoint, the band structure is not generally real. On the other hand, the PT symmetry condition (2) provides useful clues on the reality of the spectrum [2,15,16,23,24]. In particular,…”

“…In standard Hermitian optics (when V (x) is real) two different FB modes, φ n (x, k), φ m (x, k ) satisfy the orthogonality relation (15) with respect to the inner product…”

Analytical solutions to a class of nonlinear Schrödinger equations with {\cal PT} -like potentials

“…In what follows we show that equation (3) also admits a family of periodic solutions residing on a PT -like complex periodic lattice [23][24][25]. To demonstrate such states, we consider the following Jacobian periodic potentials:…”

A complex periodic QES potential and exceptional points

“…with ξ ∈ R and N being an integer, which has applications in one-dimensional crystal problems [9]. We shall be interested here in π -periodic or π -anti-periodic solutions of the corresponding Schrödinger equation, which are counterparts of the band edge wavefunctions (often called eigenfunctions) of theories for real periodic potentials (see, for example, [10]).…”