Fibonacci Potential: Structure & Spectral Features

• Fibonacci potential is generated by concatenating elementary blocks following the Fibonacci substitution rule, resulting in a nonperiodic, quasi-crystalline order.
• Spectral analysis reveals Cantor-like sets, mini-gap hierarchies, and scale-recursive fractal structures through transfer-matrix and trace map techniques.
• The concept is versatile, featuring in continuum Schrödinger operators, effective polariton models, and golden-periodic complex potentials in planar hydrodynamics.

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="(Fillman et al., 2017)", max_results=5, sort_by="relevance") arxiv_search(query="(Tanese et al., 2013)", max_results=5, sort_by="relevance") arxiv_search(query="(Pashaev, 2021)", 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 (Fillman et al., 2017, Tanese et al., 2013, Pashaev, 2021). 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 $S(b)=a$0. The dynamical system $S(b)=a$1 is strictly ergodic and minimal (Fillman et al., 2017).

For $S(b)=a$2, the real-line potential is defined by concatenation. Writing

$S(b)=a$3

one sets

$S(b)=a$4

An equivalent box notation is

$S(b)=a$5

where the box marks the location of the origin $S(b)=a$6 (Fillman et al., 2017).

A related, but not identical, realization appears in the polariton work. There the letters are denoted $S(b)=a$7 and $S(b)=a$8, with substitution rules $S(b)=a$9, $\mathbb{R}$0, or equivalently

$\mathbb{R}$1

The length of $\mathbb{R}$2 is the Fibonacci number $\mathbb{R}$3, satisfying $\mathbb{R}$4, and $\mathbb{R}$5. In the discrete description, the $\mathbb{R}$6th segment of length $\mathbb{R}$7 carries potential

$\mathbb{R}$8

In the continuous reduction used for wire-width modulation, the effective one-dimensional potential is

$\mathbb{R}$9

where $A=\{a,b\}$0 alternates between $A=\{a,b\}$1 and $A=\{a,b\}$2 in Fibonacci order (Tanese et al., 2013).

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 $A=\{a,b\}$3, the continuum Fibonacci Schrödinger operator acts on $A=\{a,b\}$4 as

$A=\{a,b\}$5

Because each $A=\{a,b\}$6, the maximal operator associated with the differential expression $A=\{a,b\}$7 with domain $A=\{a,b\}$8 is essentially self-adjoint. Hence

$A=\{a,b\}$9

and no boundary-condition choices are needed at $l_a,l_b>0$0 (Fillman et al., 2017).

Minimality and unique ergodicity imply the existence of a compact set $l_a,l_b>0$1, called the common spectrum, such that

$l_a,l_b>0$2

Moreover, $l_a,l_b>0$3 is a Cantor set of zero Lebesgue measure whenever $l_a,l_b>0$4 is aperiodic, meaning not both periodic combinations (Fillman et al., 2017).

The polariton model is formulated through an effective one-dimensional Schrödinger equation for the lower-polariton field $l_a,l_b>0$5: $l_a,l_b>0$6 where $l_a,l_b>0$7 is the polariton effective mass $l_a,l_b>0$8, $l_a,l_b>0$9 is the long axis of the wire, and Dirichlet boundary conditions $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$0 mimic the vanishing field at the etched edges (Tanese et al., 2013).

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 (Tanese et al., 2013). 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 $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$1 in specific asymptotic regimes (Fillman et al., 2017).

3. Trace map, invariant surfaces, and spectral characterization

The fine structure of the continuum spectrum is analyzed through transfer matrices. For each symbol in $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$2, one considers the fundamental solutions of

$f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$3

on the appropriate interval, with Dirichlet or Neumann data at $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$4. Denoting them by $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$5 and $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$6, the $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$7 transfer matrix is

$f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$8

Since $f_a\in L^2([0,l_a)),\qquad f_b\in L^2([0,l_b)).$9, one has

$S(a)=ab,\qquad S(b)=a.$0

This defines the curve of initial conditions

$S(a)=ab,\qquad S(b)=a.$1

for the trace map

$S(a)=ab,\qquad S(b)=a.$2

(Fillman et al., 2017).

The trace map preserves the Fricke–Vogt invariant

$S(a)=ab,\qquad S(b)=a.$3

Therefore each forward orbit $S(a)=ab,\qquad S(b)=a.$4 lies on the surface

$S(a)=ab,\qquad S(b)=a.$5

A theorem of Damanik–Fillman–Gorodetski gives the exact spectral criterion

$S(a)=ab,\qquad S(b)=a.$6

On the spectrum, $S(a)=ab,\qquad S(b)=a.$7, and the local fractal properties of $S(a)=ab,\qquad S(b)=a.$8 at energy $S(a)=ab,\qquad S(b)=a.$9 are governed by $a$0 (Fillman et al., 2017).

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 $a$1 is not merely diagnostic; it parametrizes the local fractal geometry of the spectrum.

4. Local Hausdorff dimension and asymptotic regimes

For each $a$2, the local Hausdorff dimension is defined by

$a$3

There exists a continuous, real-analytic mapping

$a$4

such that

$a$5

Its asymptotics are

$a$6

and

$a$7

(Fillman et al., 2017).

The continuum theory establishes two asymptotic regimes in which $a$8, independently of the precise shapes of $a$9 and $u=S(u)=abaababa\cdots\in A^{\mathbb N}.$0. In the high-energy regime,

$u=S(u)=abaababa\cdots\in A^{\mathbb N}.$1

and consequently

$u=S(u)=abaababa\cdots\in A^{\mathbb N}.$2

In the small-coupling regime, for

$u=S(u)=abaababa\cdots\in A^{\mathbb N}.$3

one has uniformly in $u=S(u)=abaababa\cdots\in A^{\mathbb N}.$4,

$u=S(u)=abaababa\cdots\in A^{\mathbb N}.$5

so that

$u=S(u)=abaababa\cdots\in A^{\mathbb N}.$6

(Fillman et al., 2017).

The proofs rely on perturbative transfer-matrix approximations. At large $u=S(u)=abaababa\cdots\in A^{\mathbb N}.$7 or small $u=S(u)=abaababa\cdots\in A^{\mathbb N}.$8, the fundamental solutions $u=S(u)=abaababa\cdots\in A^{\mathbb N}.$9 are close to the free solutions $\Omega_F\subset A^{\mathbb Z}$0 and $\Omega_F\subset A^{\mathbb Z}$1, leading to the estimates

$\Omega_F\subset A^{\mathbb Z}$2

$\Omega_F\subset A^{\mathbb Z}$3

and hence

$\Omega_F\subset A^{\mathbb Z}$4

Substituting these into

$\Omega_F\subset A^{\mathbb Z}$5

shows that in the free case $\Omega_F\subset A^{\mathbb Z}$6, while perturbative errors yield $\Omega_F\subset A^{\mathbb Z}$7 or $\Omega_F\subset A^{\mathbb Z}$8 (Fillman et al., 2017).

The function $\Omega_F\subset A^{\mathbb Z}$9 arises from a uniformly hyperbolic repeller on $S(b)=a$00, whose unstable-manifold contraction rates depend smoothly on $S(b)=a$01. Analytic perturbation theory gives the expansion $S(b)=a$02 as $S(b)=a$03 (Fillman et al., 2017). 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

$S(b)=a$04

About any reference energy $S(b)=a$05, one has the discrete scaling relation

$S(b)=a$06

with scaling factors $S(b)=a$07 depending on $S(b)=a$08. Equivalently,

$S(b)=a$09

where the scaling function $S(b)=a$10 is periodic of period $S(b)=a$11. This gives power-law envelopes with exponent $S(b)=a$12, modulated by log-periodic oscillations (Tanese et al., 2013).

A leading-order equivalent form is

$S(b)=a$13

which explicitly exhibits log-periodic modulation of period $S(b)=a$14 in $S(b)=a$15 (Tanese et al., 2013).

The same work states the gap-labeling theorem for one-dimensional quasi-periodic Schrödinger operators: $S(b)=a$16 with $S(b)=a$17. Equivalently, the Fourier expansion of $S(b)=a$18 has Bragg peaks at

$S(b)=a$19

and each peak opens a mini-gap whose integrated density of states is exactly $S(b)=a$20 modulo $S(b)=a$21 (Tanese et al., 2013).

In weak-potential perturbation theory, each Bragg peak $S(b)=a$22 couples degenerate plane waves $S(b)=a$23, opening a gap of width

$S(b)=a$24

The Fourier or structure-factor form of the potential is

$S(b)=a$25

(Tanese et al., 2013).

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 $S(b)=a$26, the $S(b)=a$27th Fibonacci divisors are

$S(b)=a$28

where the Fibonacci numbers satisfy $S(b)=a$29, $S(b)=a$30, $S(b)=a$31, and Binet’s formula

$S(b)=a$32

The $S(b)=a$33th Golden derivative is defined by

$S(b)=a$34

and satisfies

$S(b)=a$35

In the limit $S(b)=a$36, this reduces to the ordinary $S(b)=a$37 (Pashaev, 2021).

The corresponding $S(b)=a$38th golden exponential is

$S(b)=a$39

with $S(b)=a$40, and it obeys

$S(b)=a$41

The golden translation operator is

$S(b)=a$42

A function is golden-periodic of order $S(b)=a$43 exactly when

$S(b)=a$44

and one has

$S(b)=a$45

(Pashaev, 2021).

In the annulus

$S(b)=a$46

the primary vortex potential is

$S(b)=a$47

which fails the no-normal-flow condition on $S(b)=a$48 and $S(b)=a$49. By the classical two-circle theorem, one adds two doubly infinite image sequences: images by inversion in the inner circle at points $S(b)=a$50, and images by inversion in the outer circle at points $S(b)=a$51, $S(b)=a$52. After summation, the closed-form complex potential is

$S(b)=a$53

and the complex velocity is

$S(b)=a$54

(Pashaev, 2021).

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.