Transmission coefficient from generalized Cantor-like potentials and its multifractality

We study the scattering problem at generalized Cantor-like potentials characterized by the expansion rate $a$ and duplication number $N$, and derive an exact formula of transmittance. It was found that the transmittance is expressed with Chebyshev polynomials of the second kind, and the multifractality of the reflectance varies depending on $a$ and $N$.

Figure 1

(a) An array of $N$ potential barriers. (b) A periodic array of potential barriers.

Figure 2

Transmittance of a wave function incident on $N$ number of equally spaced potentials. (a) $N=4$. (b) $N=32$. (c) $N=264$. (d) $N=1024$. The other parameters are $L=1$, $\mathrm{\Delta }=1$, $U_0=100$.

Figure 3

Schematic diagram of the Cantor-like potentials in the $n=\mathrm{zeroth}$, first, and second stage. (a) 3-adic Cantor-like potential barrier. (b) $(2N-1)$-adic SV Cantor-like potential barrier for $a=4$, $N=2$. (c) $(2N-1)$-adic SV Cantor-like potential barrier for $a=4$, $N=3$.

Figure 4

Transmittance for the tenth stage potential $U\left(x\right)={(a/N)}^n{U}_0$ at $L=1$, $U_0=512$. (a) $a=3$, $N=2$. (b) $a=4$, $N=2$. (c) $a=4$, $N=3$. (d) $a=5$, $N=2$. (e) $a=5$, $N=3$. (f) $a=5$, $N=4$.

Figure 5

Enlarged view of transmittance peak at $\varepsilon /U_0<1$. (a) $a=3$, $N=2$. (b) $a=4$, $N=2$. (c) $a=4$, $N=3$. (d) $a=5$, $N=2$. (e) $a=5$, $N=3$. (f) $a=5$, $N=4$.

Figure 6

(a) Cantor-like potential with randomness in the heights of elemental potential walls. (b) Cantor-like potential with randomness in the positions of elemental potential walls.

Figure 7

Transmittance for the tenth stage potential $U\left(x\right)={(a/N)}^n{U}_0$ at $a=4$, $N=3$, $L=1$, $U_0=512$. The left-hand side shows the transmittance near the wave number showing the resonant tunnel. The right-hand side shows the transmittance for $\varepsilon /U_0>1$. (a) Results for Eq. (21). (b) Results for Eq. (29) at $p_1=0.1$. (c) Results for Eq. (30).

Figure 8

Cantor-like potential where the potential height is not constant and the numerical calculation result is at $L=1$, $U=512$, $n=10$, $a=4$. (a) Schematic diagram of the Cantor-like potential in the $n=\text{zeroth}$, first, and second, where the potential height is not constant. (b) Numerical result when the potential height is constant. (c) Numerical result when the potential height is not constant.

Figure 9

$G\left(k\right)$ in a semilogarithmic scale of $k$. (a) $a=3$, $N=2$. (b) $a=4$, $N=2$. (c) $a=4$, $N=3$. (d) $a=5$, $N=2$. (e) $a=5$, $N=3$. (f) $a=5$, $N=4$.

Figure 10

(a) $f\left(\alpha \right)$ spectra at $a=3$, 4, and 5 for a fixed value of $N=2$. The solid line is $a=3$, the dashed line is $a=4$, and the dotted line is $a=5$. (b) $f\left(\alpha \right)$ spectra at $N=2$, 4, and 5 for a fixed value of $a=5$. The solid line is $N=4$, the dashed line is $N=3$, and the dotted line is $N=2$.