Quantum chaos in one dimension?

In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest $N$ eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit $N\to \infty$ the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.

Figure 1

The potentials calculated with the Wu method (solid line) and the dressing transformation (dashed line) for $N=50$ are shown. The potentials provided by these methods are almost identical in the [$-$10, 10] range. Inset shows the same potentials in the [0, 3] interval.

Figure 2

The potential calculated with the Wu method for $N=5$ (bottom), $50$ (middle), and $250$ (top). The graphs for $N=50$ and $N=250$ are shifted vertically.

Figure 3

(a) The potential calculated with the Wu method for $N=5$ (continuous line), $N=50$ (dashed line), and $N=250$ (points) in the interval $[0;2]$. (b) The meaning of $\Delta x$ and $\Delta y$.

Figure 4

The average distance of two adjacent extrema $\stackrel{̲}{\Delta x}$ is plotted as a function of $N$ for both methods. The solid and dashed lines are plotted only to guide the eye and are obtained by fitting the function $A+{\mathit{BN}}^{-C}$ onto the data. The inset is the same figure on a log-log scale.

Figure 5

The average vertical “jump” between two adjacent extrema, $\stackrel{̲}{\Delta y}$, is depicted as a function of $N$ obtained using both methods.

Figure 6

The histogram of the parameter $c$ is depicted for different values of $N$.

Figure 7

The average of $c$ as a function of $N$ with the Wu method and the dressing transformation.

Figure 8

The Fourier transform of $V_{\mathrm{fl}}(x)$ at $N=5$ (upper graph) and at $N=250$ (lower graph). $\mathcal{F}[V_{\mathrm{fl}}(x)]$ gets wider as $N$ increases. $\mathcal{F}[V_{\mathrm{fl}}(x)]$ vanishes beyond a characteristic wave number for every $N$ denoted as $f_{\max }$. Inset: $f_{max}$ as a function of $N$ with both methods. The two curves show very similar behavior.

Figure 9

The box dimension of $V_{\mathrm{fl}}(x)$ as a function of $N$. Inset: $V_{\mathrm{fl}}(x)$ at $N=250$.

Figure 10

The number of nodes of the potential is plotted as a function of $N$. Inset shows the absolute value of $V_{\mathrm{fl}}(x)$ integrated over the entire $x$ axis. The solid and dashed lines correspond the same method in both graphs.