Fractal dimensions of a one-dimensional local potential model

Question

In the paper Riemann zero and a fractal potential by Wu and Sprung, they found that the nontrivial Riemann zero are reproduced using a one-dimensional local potential model.

They then describe the potential has having ‘a fractal structure of dimension $d=1.5$.

What does a fractal dimension mean? What is the physical realisation of the this in terms of quantum mechanics.

Answer 1

There are multiple ways of generalizing the concept of dimension, especially in topology using boundaries or open coverings. A very intuitive generalization is the Lebesgue dimension defined as the minimal number $n$, so that for every open covering, there is a refinement, so that every point is in at most $n+1$ covering sets. Imagine it in $\mathbb{R}$, $\mathbb{R}^2$ or $\mathbb{R}^3$, where every open covering must have an intersection of at least two, three or four covering sets respectively. This is mentioned in the paper as "box-counting method". Those different dimensions do not have to be equal, an example is given in https://doi.org/10.2969/jmsj/01820158. For metric spaces, a suitable generalization is the Hausdorff dimension, often also called the fractal dimension.

For vector spaces, we define the dimension as the well-defined number of vectors of a basis, which is the number of independent directions. This makes the dimension a natural number of course. Let us now take a different approach by using a real number $a>0$ to stretch our figures.

1. Apply it to a one-dimensional line and you get $a$ of the of the original ones.
2. Apply it to a two-dimensional square and you get $a^2$ of the original ones.
3. Apply it to a three-dimensional cube and you get $a^3$ of the original ones.

The exponent is exactly the dimension, which should not be a surprise as for every direction we have, we get one stretching factor. The surprise is applying this method to fractals, as their fractal nature results in different natural values than powers of two.

1. Take $a=3$ for the Koch curve, then you get $3^d\stackrel{!}{=}4$ of the original ones, so its dimension is $d=\log_3(4)\approx 1.262$.
2. Take $a=2$ for the Sierpiński triangle, then you get $2^d\stackrel{!}{=}3$ of the original ones, so its dimension is $d=\log_2(3)\approx 1.585$.
3. Take $a=3$ for the Menger sponge, then you get $3^d\stackrel{!}{=}20$ of the original ones, so its dimension is $d=\log_3(20)\approx 2.727$

As you see, we get logarithms, which appear multiple times in the paper. But the calculation I described only works because fractals are regular structures. The Hausdorff dimension can be generalized using the Hausdorff measure making use of an open covering described in https://en.wikipedia.org/wiki/Hausdorff_measure. Now we can get started getting crazy results. The coastlines of Ireland, Great Britain and Norway have dimension 1.22, 1.25 and 1.52 respectively. The surface of Broccoli and Cauliflower have 2.7 and 2.8 while your brain and your lungs have 2.79 and 2.97 respectively. This shows that the fractal dimension gives us an impression of the amount of convolutions and folds contributing to its complexity.

In the paper, the scalar potential $V$ is only depended on one variable, therefore we expect the fractal dimension of its graph to be between $1$ and $2$. Looking at FIG. 2 you can see all the twists and turns of the graph, oscillating more and faster the more zeros of the Riemann $\zeta$ function you consider. As a result, the graph has a dimension of roughly $1.5$, but the paper gives no physical interpretation for it. As far as I know, there are no applications of fractal dimension in quantum mechanics as dimension in general is used as an invariant to classify spaces, which is a purely mathematical problem.