Superoscillatory $\mathcal{PT}$-symmetric potentials

We introduce the one-dimensional $\mathcal{PT}$-symmetric Schrödinger equation, with complex potentials in the form of the canonical superoscillatory and suboscillatory functions known in quantum mechanics and optics. While the suboscillatorylike potential always generates an entirely real eigenvalue spectrum, its counterpart based on the superoscillatory wave function gives rise to an intricate pattern of $\mathcal{PT}$-symmetry-breaking transitions, controlled by the parameters of the superoscillatory function. One scenario of the transitions proceeds smoothly via a set of threshold values, while another one exhibits a sudden jump to the broken $\mathcal{PT}$ symmetry. Another noteworthy finding is the possibility of restoration of the $\mathcal{PT}$ symmetry, following its original loss, in the course of the variation of the parameters.

Figure 1

Energy-band structure (right) of potential ${\mathbf{V}}_{\mathbf{SO}}\left(\xi \right)$ (left), calculated for $\mathbf{N}=\mathbf{4},\mathbf{a}=\mathbf{2}$. A black arrow marks the location where $E=N^2$.

Figure 2

$\mathcal{PT}$-symmetry breaking measure ${\rho }_{\mathbf{1}}(\mathbf{a},\mathbf{N})$. The darkest (brightest) color, corresponding to ${\rho }_1(a,N)=0 [{\rho }_1(a,N)=1]$, shows the domain of parameters where the first band is entirely real (complex). Some specific cases of interest are marked, including positive integer $N$ at $a=1$, where potential (7) is $exp\left(iN\xi \right)$ (white crosses), even integer $N$ (white dotted lines), $N=1$ (the white dashed dotted line), and, finally, $N=4,a=2.5$ (black arrows), where a sharp $\mathcal{PT}$-symmetry transition is clearly observed in the lower subframe zooming this region.

Figure 3

Eigenvalue-band structure for selected potentials. Here, $a=2.5$, while (a) $N=3.9$, (b) $N=4.0$, and (c) $N=4.1$. Left column: the real (blue line) and imaginary (red dashed line) parts of each potential. Right column: the band diagrams showing the real and imaginary parts of the eigenvalues in the first and second bands.

Figure 4

Calculated discriminant function $\mathrm{\Delta }\left(\mathbf{E}\right)$, which determines the existence of Bloch wave functions corresponding to real eigenvalues, for $a=2.5$ and (a) $N=3.9$, (b) $N=4.0$, and (c) $N=4.1$. Dotted vertical lines mark the boundaries $\mathrm{\Delta }\left(E\right)=\pm 2$ indicating the existence area for the wave functions. The arrow marks a sharp minimum at $E\approx 15.56$.

Figure 5

Simulated evolution of the local intensity of the broad initial Gaussian, displayed for potential (7) with $a=2.5$ and $N=3.9$ (left), $N=4.0$ (center), and $N=4.1$ (right). In the left and right panels, the log scale is used.