---
title: 'Fibonacci Potential: Structure & Spectral Features'
url: https://www.emergentmind.com/topics/fibonacci-potential
type: topic
---

# Fibonacci Potential: Structure & Spectral Features

Searching arXiv for papers on Fibonacci potentials and the provided IDs.
arxiv_search(query="Fibonacci potential Schrödinger Fibonacci continuum spectrum polariton", max_results=10, sort_by="relevance")
arxiv_search(query="1702.04337", max_results=5, sort_by="relevance")
arxiv_search(query="1311.3453", max_results=5, sort_by="relevance")
arxiv_search(query="2108.10759", max_results=5, sort_by="relevance")
A Fibonacci potential is a potential generated by concatenating two elementary constituents according to the Fibonacci substitution rule \(S(a)=ab\), \(S(b)=a\). In the cited literature, this construction appears in several technically distinct settings: continuum Schrödinger operators on \(\mathbb{R}\) whose local potential pieces are arranged by the Fibonacci subshift, effective one-dimensional potentials for polariton gases in Fibonacci quasi-periodic cavities, and golden-periodic complex potentials arising from the method of images in planar hydrodynamics [1702.04337] [1311.3453] [2108.10759]. Across these settings, the common structural feature is substitutional order without periodicity, which yields Cantor-like spectra, mini-gap hierarchies, and scale-recursive analytic structures.

## 1. Substitutional definition and geometric realization

The underlying symbolic construction starts from a two-letter alphabet. In the continuum Schrödinger setting, one takes \(A=\{a,b\}\), fixes two positive lengths \(l_a,l_b>0\), and chooses two building-block potentials
\[
f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).
\]
The Fibonacci substitution is
\[
S(a)=ab,\qquad S(b)=a.
\]
Starting from \(a\) and iterating, one obtains the one-sided fixed point
\[
u=S(u)=abaababa\cdots\in A^{\mathbb N}.
\]
The associated two-sided Fibonacci subshift \(\Omega_F\subset A^{\mathbb Z}\) consists of all bi-infinite words whose every finite subword appears somewhere in \(u\). The dynamical system \((\Omega_F,\text{shift})\) is strictly ergodic and minimal [1702.04337].

For \(\omega\in\Omega_F\), the real-line potential is defined by concatenation. Writing
\[
S_n(\omega)=\sum_{k<n} l_{\omega_k},\qquad n\in\mathbb Z,
\]
one sets
\[
V_\omega(x)=f_{\omega_n}(x-S_n(\omega))
\quad\text{whenever}\quad
x\in [S_n(\omega),S_{n+1}(\omega)).
\]
An equivalent box notation is
\[
V_\omega=\cdots|\,f_{\omega_{-1}}\,|\,f_{\omega_0}\,|\,f_{\omega_1}\,|\cdots,
\]
where the box marks the location of the origin \(x=0\) [1702.04337].

A related, but not identical, realization appears in the polariton work. There the letters are denoted \(A\) and \(B\), with substitution rules \(A\to AB\), \(B\to A\), or equivalently
\[
S_1=B,\qquad S_2=A,\qquad S_j=S_{j-2}S_{j-1}\quad (j\ge 3).
\]
The length of \(S_j\) is the Fibonacci number \(F_j\), satisfying \(F_{j+1}=F_j+F_{j-1}\), and \(F_{j+1}/F_j\to \phi\equiv (1+\sqrt5)/2\). In the discrete description, the \(n\)th segment of length \(a\) carries potential
\[
V_n=
\begin{cases}
V_A,&\text{if the \(n\)th letter is A},\\
V_B,&\text{if the \(n\)th letter is B}.
\end{cases}
\]
In the continuous reduction used for wire-width modulation, the effective one-dimensional potential is
\[
V(x)=\frac{\pi^2}{w(x)^2}+\frac{\pi^2+3}{12}\Bigl(\frac{w'(x)}{w(x)}\Bigr)^2,
\]
where \(w(x)\) alternates between \(w_1=w_A\) and \(w_2=w_B\) in Fibonacci order [1311.3453].

These constructions show that a Fibonacci potential is not restricted to a single formalism. It may be implemented as a continuum concatenation of compactly supported pieces or as an effective quasiperiodic profile derived from geometry, but in both cases the symbolic order is generated by the same inflation rule.

## 2. Continuum Fibonacci Schrödinger operators

For each \(\omega\in\Omega_F\), the continuum Fibonacci Schrödinger operator acts on \(L^2(\mathbb R)\) as
\[
H_\omega=-\frac{d^2}{dx^2}+V_\omega(x).
\]
Because each \(V_\omega\in L^\infty_{\mathrm{loc}}(\mathbb R)\), the maximal operator associated with the differential expression \(-d^2/dx^2+V_\omega\) with domain \(H^2(\mathbb R)\) is essentially self-adjoint. Hence
\[
\mathrm{Dom}(H_\omega)=H^2(\mathbb R),
\]
and no boundary-condition choices are needed at \(\pm\infty\) [1702.04337].

Minimality and unique ergodicity imply the existence of a compact set \(\Sigma\subset\mathbb R\), called the common spectrum, such that
\[
\sigma(H_\omega)=\Sigma
\qquad\text{for every }\omega\in\Omega_F.
\]
Moreover, \(\Sigma\) is a Cantor set of zero Lebesgue measure whenever \((f_a,f_b)\) is aperiodic, meaning not both periodic combinations [1702.04337].

The polariton model is formulated through an effective one-dimensional Schrödinger equation for the lower-polariton field \(\psi(x)\):
\[
-\frac{\hbar^2}{2m_p}\frac{d^2\psi}{dx^2}+V(x)\psi(x)=E\psi(x),
\]
where \(m_p\) is the polariton effective mass \((\simeq 10^{-5}\text{ electron mass})\), \(x\in[0,L]\) is the long axis of the wire, and Dirichlet boundary conditions \(\psi(0)=\psi(L)=0\) mimic the vanishing field at the etched edges [1311.3453].

In the infinite Fibonacci limit, the spectral type in that setting is described as neither purely continuous nor pure point but singular continuous, that is, a Cantor-like set of zero total Lebesgue measure [1311.3453]. A common misconception is that “almost full-dimensional” fractal spectra should become interval-like. The continuum theory shows otherwise: for aperiodic building blocks the spectrum remains a Cantor set of zero Lebesgue measure, even though the local Hausdorff dimension approaches \(1\) in specific asymptotic regimes [1702.04337].

## 3. Trace map, invariant surfaces, and spectral characterization

The fine structure of the continuum spectrum is analyzed through transfer matrices. For each symbol in \(\{a,b,ab\}\), one considers the fundamental solutions of
\[
-y''(x)+f_a(x)y(x)=Ey(x)
\]
on the appropriate interval, with Dirichlet or Neumann data at \(x=0\). Denoting them by \(v_{a,D}(\cdot,E)\) and \(v_{a,N}(\cdot,E)\), the \(2\times 2\) transfer matrix is
\[
M_a(E)=
\begin{pmatrix}
v_{a,N}(l_a,E)&v_{a,D}(l_a,E)\\
v'_{a,N}(l_a,E)&v'_{a,D}(l_a,E)
\end{pmatrix},
\qquad
x_a(E)=\tfrac12\mathrm{Tr}\,M_a(E).
\]
Since \(S(ab)=ba\), one has
\[
M_{ab}(E)=M_b(E)M_a(E),
\qquad
x_{ab}(E)=\tfrac12\mathrm{Tr}(M_b(E)M_a(E)).
\]
This defines the curve of initial conditions
\[
y(E)=\bigl(x_{ab}(E),x_a(E),x_b(E)\bigr)\in\mathbb R^3
\]
for the trace map
\[
T(x,y,z)=(2xy-z,\;x,\;y)
\]
[1702.04337].

The trace map preserves the Fricke–Vogt invariant
\[
I(x,y,z)=x^2+y^2+z^2-2xyz-1.
\]
Therefore each forward orbit \(\{T^n(y(E)):n\ge 0\}\) lies on the surface
\[
S_{I(E)}=\{(x,y,z): I(x,y,z)=I(E)\}.
\]
A theorem of Damanik–Fillman–Gorodetski gives the exact spectral criterion
\[
E\in\Sigma
\iff
\{T^n(y(E)):n\ge 0\}\ \text{is bounded}.
\]
On the spectrum, \(I(E)\ge 0\), and the local fractal properties of \(\Sigma\) at energy \(E\) are governed by \(I(E)\) [1702.04337].

This characterization is central because it converts a spectral problem for a continuum quasiperiodic operator into a dynamical problem on invariant surfaces. The role of \(I(E)\) is not merely diagnostic; it parametrizes the local fractal geometry of the spectrum.

## 4. Local Hausdorff dimension and asymptotic regimes

For each \(E\in\Sigma\), the local Hausdorff dimension is defined by
\[
\dim_H(\Sigma;E)=\lim_{\varepsilon\to 0}\dim_H(\Sigma\cap(E-\varepsilon,E+\varepsilon)).
\]
There exists a continuous, real-analytic mapping
\[
D:[0,\infty)\to(0,1],\qquad D(0)=1,
\]
such that
\[
\dim_H(\Sigma;E)=D\bigl(I(E)\bigr).
\]
Its asymptotics are
\[
1-D(I)=O(\sqrt I)\qquad (I\to 0^+),
\]
and
\[
D(I)=\frac{2\log(1+\sqrt2)}{\log I}+o\!\bigl(1/\log I\bigr)
\qquad (I\to\infty)
\]
[1702.04337].

The continuum theory establishes two asymptotic regimes in which \(I(E)\to 0\), independently of the precise shapes of \(f_a\) and \(f_b\). In the high-energy regime,
\[
I(E)=O(E^{-1/2})
\qquad (E\to\infty),
\]
and consequently
\[
\lim_{E\to\infty}\inf_{E'\ge E}\dim_H(\Sigma;E')=1.
\]
In the small-coupling regime, for
\[
H_{\omega,\lambda}=-\frac{d^2}{dx^2}+\lambda V_\omega(x),
\]
one has uniformly in \(E\in\Sigma\),
\[
I(E,\lambda)=O(\lambda)\qquad (\lambda\to 0),
\]
so that
\[
\lim_{\lambda\to 0}\inf_{E'\in\Sigma}\dim_H(\Sigma_\lambda;E')=1
\]
[1702.04337].

The proofs rely on perturbative transfer-matrix approximations. At large \(E\) or small \(\lambda\), the fundamental solutions \(v_{a,D/N}(x,E,\lambda)\) are close to the free solutions \(\cos(\sqrt E\,x)\) and \(\sin(\sqrt E\,x)/\sqrt E\), leading to the estimates
\[
M_a(E,\lambda)=M_a^0(E)+O(E^{-1/2}) \quad\text{(high \(E\))},
\]
\[
M_a(E,\lambda)=I+O(\lambda) \quad\text{(small \(\lambda\))},
\]
and hence
\[
x_a(E,\lambda)=\cos(\sqrt E\,l_a)+o(1).
\]
Substituting these into
\[
I(E)=x_{ab}^2+x_a^2+x_b^2-2x_{ab}x_ax_b-1
\]
shows that in the free case \(I\equiv 0\), while perturbative errors yield \(I=O(E^{-1/2})\) or \(O(\lambda)\) [1702.04337].

The function \(D(I)\) arises from a uniformly hyperbolic repeller on \(S_I\), whose unstable-manifold contraction rates depend smoothly on \(I\). Analytic perturbation theory gives the expansion \(D(I)=1-c\sqrt I+\cdots\) as \(I\to 0\) [1702.04337]. This makes precise the sense in which the continuum Fibonacci spectrum becomes almost full-dimensional in Hausdorff dimension while retaining its Cantor character.

## 5. Fractal energy spectrum, self-similarity, and gap labeling

In the polariton realization, the fractal character of a Fibonacci potential is described through the integrated density of states
\[
\mu(\varepsilon)=\int_{-\infty}^{\varepsilon}\rho(\varepsilon')\,d\varepsilon'.
\]
About any reference energy \(\varepsilon_u\), one has the discrete scaling relation
\[
\mu(\varepsilon_u+\Delta\varepsilon)-\mu(\varepsilon_u)
=
\frac{1}{\alpha}
\Bigl[
\mu(\varepsilon_u+\beta\,\Delta\varepsilon)-\mu(\varepsilon_u)
\Bigr],
\]
with scaling factors \(\alpha,\beta\) depending on \(\varepsilon_u\). Equivalently,
\[
\mathcal N_{\varepsilon_u}(\varepsilon)
\equiv
\mu(\varepsilon)-\mu(\varepsilon_u)
=
|\varepsilon-\varepsilon_u|^\gamma
\mathcal F\!\biggl(
\frac{\ln|\varepsilon-\varepsilon_u|}{\ln\beta}
\biggr),
\qquad
\gamma=\frac{\ln\alpha}{\ln\beta},
\]
where the scaling function \(\mathcal F(z)\) is periodic of period \(1\). This gives power-law envelopes with exponent \(\gamma\in(0,1)\), modulated by log-periodic oscillations [1311.3453].

A leading-order equivalent form is
\[
\ln \mathcal N
\approx
\gamma\ln|\Delta\varepsilon|
+
A\cos\!\Bigl[
2\pi \frac{\ln|\Delta\varepsilon|}{\ln\beta}+\varphi_0
\Bigr]
+\cdots,
\]
which explicitly exhibits log-periodic modulation of period \(\ln\beta\) in \(\ln|\Delta\varepsilon|\) [1311.3453].

The same work states the gap-labeling theorem for one-dimensional quasi-periodic Schrödinger operators:
\[
N(E_{\rm gap})\equiv \mu(E_{\rm gap})
=
p+q\,\phi^{-1}
\quad(\bmod\,1),
\]
with \(p,q\in\mathbb Z\). Equivalently, the Fourier expansion of \(V(x)\) has Bragg peaks at
\[
Q_{p,q}=\frac{2\pi}{a}\bigl(p+q\,\phi^{-1}\bigr),
\]
and each peak opens a mini-gap whose integrated density of states is exactly \(p+q\,\phi^{-1}\) modulo \(1\) [1311.3453].

In weak-potential perturbation theory, each Bragg peak \(\chi_q\) couples degenerate plane waves \(\pm Q_{p,q}/2\), opening a gap of width
\[
\Delta_{p,q}=2|V_q|
\quad\text{at energy}\quad
E=E_{Q_{p,q}/2}.
\]
The Fourier or structure-factor form of the potential is
\[
V(x)=\sum_{n\in\mathbb Z}\chi(\phi^{-1}n)\,u_b(x-na),
\qquad
V(k)=\widetilde u_b(k)\sum_{p,q}\chi_q\,\delta\bigl(ka-2\pi(p+q\phi^{-1})\bigr)
\]
[1311.3453].

These formulas sharpen the spectral meaning of Fibonacci order. The spectrum is not merely fragmented; its gaps are topologically indexed, and its integrated density of states displays discrete self-similarity with log-periodic corrections.

## 6. Golden-periodic complex potentials in planar hydrodynamics

A distinct usage of Fibonacci structure occurs in planar hydrodynamics through the PQ-calculus of Fibonacci divisors. For any fixed integer \(k\ne 0\), the \(k\)th Fibonacci divisors are
\[
F_n^{(k)}:=\frac{F_{kn}}{F_k},
\]
where the Fibonacci numbers satisfy \(F_0=0\), \(F_1=1\), \(F_{n+1}=F_n+F_{n-1}\), and Binet’s formula
\[
F_n=\frac{\phi^n-(-\phi)^{-n}}{\sqrt5}.
\]
The \(k\)th Golden derivative is defined by
\[
D_F^{(k)}[f](z)=
\frac{f(\phi^k z)-f((-\phi)^{-k}z)}
{(\phi^k-(-\phi)^{-k})z},
\]
and satisfies
\[
D_F^{(k)}[z^n]=F_n^{(k)}z^{n-1}.
\]
In the limit \(k\to 0\), this reduces to the ordinary \(d/dz\) [2108.10759].

The corresponding \(k\)th golden exponential is
\[
E_F^{(k)}(z):=\sum_{n=0}^{\infty}\frac{z^n}{F_n^{(k)}!},
\]
with \(F_n^{(k)}!:=F_1^{(k)}F_2^{(k)}\cdots F_n^{(k)}\), and it obeys
\[
D_F^{(k)}E_F^{(k)}(\lambda z)=\lambda E_F^{(k)}(\lambda z).
\]
The golden translation operator is
\[
T_F^{(k)}(a):=\exp_F^{(k)}\bigl(aD_F^{(k)}\bigr)
=
\sum_{m=0}^{\infty}\frac{a^m(D_F^{(k)})^m}{F_m^{(k)}!}.
\]
A function is golden-periodic of order \(k\) exactly when
\[
F(\phi^k z)=F(z),
\]
and one has
\[
D_F^{(k)}[F]=0 \iff F(\phi^k z)=F(z)
\]
[2108.10759].

In the annulus
\[
A_m=\{z:1<|z|<\phi^m\},
\]
the primary vortex potential is
\[
W_0(z)=\frac{i\Gamma}{2\pi}\ln(z-z_0),
\]
which fails the no-normal-flow condition on \(|z|=1\) and \(|z|=\phi^k\). By the classical two-circle theorem, one adds two doubly infinite image sequences: images by inversion in the inner circle at points \((1/\overline z_0)\phi^{-kn}\), and images by inversion in the outer circle at points \((\phi^k/\overline z_0)\phi^{kn}\), \(n\in\mathbb Z\). After summation, the closed-form complex potential is
\[
W_F(z)=\frac{i\Gamma}{2\pi}
\sum_{n=-\infty}^{\infty}
\Bigl[
\ln\bigl(z-\phi^{kn}z_0\bigr)
-
\ln\bigl(z-\phi^{kn}\phi^k/\overline z_0\bigr)
\Bigr],
\]
and the complex velocity is
\[
V(z)=\frac{dW_F}{dz}
=
\frac{i\Gamma}{2\pi}
\sum_{n=-\infty}^{\infty}
\Bigl[
\frac{1}{z-\phi^{kn}z_0}
-
\frac{1}{z-\phi^{kn}\phi^k/\overline z_0}
\Bigr]
\]
[2108.10759].

This hydrodynamic usage differs from the Schrödinger and polariton settings: the word “potential” here denotes a complex potential rather than a scalar potential in an eigenvalue problem. The shared element is golden-periodic recursion. A plausible implication is that Fibonacci order functions as a transferable analytic template across spectral theory and image-based boundary constructions, even when the governing equations are different.

Source: https://www.emergentmind.com/topics/fibonacci-potential