Period Doubling HamiltonianThe Spectrum
Methods and Results
Asymptotic Analysis of the Spectrum of theDiscrete Hamiltonian with Period Doubling
Potential
Meg Fields, Tara Hudson, Maria Markovich
Cornell Summer Math Institute 2013
August 2, 2013
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
MotivationQuasi-Crystals
The discovery of quasi-crystals in the 1980’s has provoked manynew questions in physics and mathematics [1].
Quasi-crystals are materials that share some properties withcrystals, but have aperiodic lattices.
Mathematically, we can model quasi-crystals using one-dimensionalHamiltonian operators with aperiodic potentials [1], [2], [5].
Fractal properties of the spectra of these operators affectquantum diffusion patterns of electrons in quasi-crystallinematerials [7], [4], [8].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
MotivationQuasi-Crystals
The discovery of quasi-crystals in the 1980’s has provoked manynew questions in physics and mathematics [1].
Quasi-crystals are materials that share some properties withcrystals, but have aperiodic lattices.
Mathematically, we can model quasi-crystals using one-dimensionalHamiltonian operators with aperiodic potentials [1], [2], [5].
Fractal properties of the spectra of these operators affectquantum diffusion patterns of electrons in quasi-crystallinematerials [7], [4], [8].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
MotivationQuasi-Crystals
The discovery of quasi-crystals in the 1980’s has provoked manynew questions in physics and mathematics [1].
Quasi-crystals are materials that share some properties withcrystals, but have aperiodic lattices.
Mathematically, we can model quasi-crystals using one-dimensionalHamiltonian operators with aperiodic potentials [1], [2], [5].
Fractal properties of the spectra of these operators affectquantum diffusion patterns of electrons in quasi-crystallinematerials [7], [4], [8].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
MotivationQuasi-Crystals
The discovery of quasi-crystals in the 1980’s has provoked manynew questions in physics and mathematics [1].
Quasi-crystals are materials that share some properties withcrystals, but have aperiodic lattices.
Mathematically, we can model quasi-crystals using one-dimensionalHamiltonian operators with aperiodic potentials [1], [2], [5].
Fractal properties of the spectra of these operators affectquantum diffusion patterns of electrons in quasi-crystallinematerials [7], [4], [8].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) =
A B A A
S3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A B
S2(A) =
A B A A
S3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B
A AS3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B A A
S3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A
A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
The Period Doubling Sequence
The period doubling sequence is given by S∞(A) whereS : A 7→ AB S : B 7→ AA
S1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B
To obtain a sequence that is infinite in both directions, we reflectthe sequence over the initial letter
...B A B A A A B A A B A A A B A B...
For computational purposes, we define A = 1 and B = −1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
Defining the Operator
The Hamiltonian operator (HV ) with period doubling potential is:
HV = −∆ + V
Where,
∆ is the discrete Laplacian of Z,
∆ =
0 −1 0 0 . . .−1 0 −1 0 . . .0 −1 0 −1 . . ....
......
.... . .
V is a multiplication operator which inputs the perioddoubling sequence into the main diagonal.
This operator acts on a sequence in `2(Z) [2].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Period Doubling SequenceDefining the Operator
Defining the Operator
The Hamiltonian operator (HV ) with period doubling potential is:
HV = −∆ + V
Where,
∆ is the discrete Laplacian of Z,
∆ =
0 −1 0 0 . . .−1 0 −1 0 . . .0 −1 0 −1 . . ....
......
.... . .
V is a multiplication operator which inputs the perioddoubling sequence into the main diagonal.
This operator acts on a sequence in `2(Z) [2].Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Methods of Approximating the Spectrum
The spectrum of an operator is a generalization of eigenvalues fora finite dimensional matrix.
To numerically approximate the spectrum of the Hamiltonian withperiod doubling potential, two methods will be employed:
1 Truncate the matrix representation of the Hamiltonian andfind the eigenvalues
2 Iterate a trace map and identify initial values which are notunstable
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Methods of Approximating the Spectrum
The spectrum of an operator is a generalization of eigenvalues fora finite dimensional matrix.
To numerically approximate the spectrum of the Hamiltonian withperiod doubling potential, two methods will be employed:
1 Truncate the matrix representation of the Hamiltonian andfind the eigenvalues
2 Iterate a trace map and identify initial values which are notunstable
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Methods of Approximating the Spectrum
The spectrum of an operator is a generalization of eigenvalues fora finite dimensional matrix.
To numerically approximate the spectrum of the Hamiltonian withperiod doubling potential, two methods will be employed:
1 Truncate the matrix representation of the Hamiltonian andfind the eigenvalues
2 Iterate a trace map and identify initial values which are notunstable
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Methods of Approximating the Spectrum
The spectrum of an operator is a generalization of eigenvalues fora finite dimensional matrix.
To numerically approximate the spectrum of the Hamiltonian withperiod doubling potential, two methods will be employed:
1 Truncate the matrix representation of the Hamiltonian andfind the eigenvalues
2 Iterate a trace map and identify initial values which are notunstable
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The trace map, is a recursively defined function developed fromthe eigenvalue equation [2].
The map describes solutions to the equation in the following way:
Initial values will be defined in terms of E, V ∈ R.
For initial points that are not unstable, E will be a value inthe spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The trace map, is a recursively defined function developed fromthe eigenvalue equation [2].
The map describes solutions to the equation in the following way:
Initial values will be defined in terms of E, V ∈ R.
For initial points that are not unstable, E will be a value inthe spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The trace map, is a recursively defined function developed fromthe eigenvalue equation [2].
The map describes solutions to the equation in the following way:
Initial values will be defined in terms of E, V ∈ R.
For initial points that are not unstable, E will be a value inthe spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The trace map, is a recursively defined function developed fromthe eigenvalue equation [2].
The map describes solutions to the equation in the following way:
Initial values will be defined in terms of E, V ∈ R.
For initial points that are not unstable, E will be a value inthe spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The initial values for the map are:
x0 := E − V and y0 := E + V ,
for some E, V ∈ R.
The trace map is,xn+1 := xnyn − 2 and yn+1 := (xn)2 − 2.
[2]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Defining the Trace Map
The initial values for the map are:
x0 := E − V and y0 := E + V ,
for some E, V ∈ R.
The trace map is,xn+1 := xnyn − 2 and yn+1 := (xn)2 − 2.
[2]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
First iterate the trace map and find the unstable points.
Definition (Unstable from [2])
A point (x, y) ∈ R2 is unstable ifx = x0 = E − V, y = y0 = E + V, and there is some N such thatfor all n > N , |xn| > 2.
Define a region
A := {(x, y) | y > 2 , |x| > 2}.
Theorem (Bellisard, Bovier, Ghez)
A point (x, y) is unstable if and only if there exists an n such that(xn, yn) ∈ A.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
First iterate the trace map and find the unstable points.
Definition (Unstable from [2])
A point (x, y) ∈ R2 is unstable ifx = x0 = E − V, y = y0 = E + V, and there is some N such thatfor all n > N , |xn| > 2.
Define a region
A := {(x, y) | y > 2 , |x| > 2}.
Theorem (Bellisard, Bovier, Ghez)
A point (x, y) is unstable if and only if there exists an n such that(xn, yn) ∈ A.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
First iterate the trace map and find the unstable points.
Definition (Unstable from [2])
A point (x, y) ∈ R2 is unstable ifx = x0 = E − V, y = y0 = E + V, and there is some N such thatfor all n > N , |xn| > 2.
Define a region
A := {(x, y) | y > 2 , |x| > 2}.
Theorem (Bellisard, Bovier, Ghez)
A point (x, y) is unstable if and only if there exists an n such that(xn, yn) ∈ A.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
E is in the spectrum of HV if and only if x = E − V andy = E + V are such that (x, y) ∈ UC .
We call V the coupling constant, a real valued parameter.
Theorem (Bellisard, Bovier, Ghez)
For a fixed coupling constant greater than zero, the spectrum ofthe Hamiltonian is a Cantor set of Lebesgue measure zero.
So it is natural to ask questions about the Hausdorff dimension,box-counting dimension, and thickness of the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
E is in the spectrum of HV if and only if x = E − V andy = E + V are such that (x, y) ∈ UC .
We call V the coupling constant, a real valued parameter.
Theorem (Bellisard, Bovier, Ghez)
For a fixed coupling constant greater than zero, the spectrum ofthe Hamiltonian is a Cantor set of Lebesgue measure zero.
So it is natural to ask questions about the Hausdorff dimension,box-counting dimension, and thickness of the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Approximating the Spectrum
Let U be the set of unstable points.
Theorem (Bellisard, Bovier, Ghez)
E is in the spectrum of HV if and only if x = E − V andy = E + V are such that (x, y) ∈ UC .
We call V the coupling constant, a real valued parameter.
Theorem (Bellisard, Bovier, Ghez)
For a fixed coupling constant greater than zero, the spectrum ofthe Hamiltonian is a Cantor set of Lebesgue measure zero.
So it is natural to ask questions about the Hausdorff dimension,box-counting dimension, and thickness of the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
y
x
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
V
E
Student Version of MATLAB
Top: A numerical approximation of UC .
Bottom: An approximation of the spectrum for a 0 ≤ V ≤ 1 [2].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
The Spectrum for a Fixed Coupling Constant
We are interested in the fractal properties of the spectrum as thecoupling constant approaches 0 [5].
For V = 0, the spectrum is the closed interval [−2, 2] ∈ R, and forV > 0, recall that the spectrum is a Cantor set.
To numerically approximate fractal measures for these sets, we usea MATLAB function crossSection to discretize the spectrum fora fixed V .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
The Spectrum for a Fixed Coupling Constant
We are interested in the fractal properties of the spectrum as thecoupling constant approaches 0 [5].
For V = 0, the spectrum is the closed interval [−2, 2] ∈ R, and forV > 0, recall that the spectrum is a Cantor set.
To numerically approximate fractal measures for these sets, we usea MATLAB function crossSection to discretize the spectrum fora fixed V .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
The Spectrum for a Fixed Coupling Constant
We are interested in the fractal properties of the spectrum as thecoupling constant approaches 0 [5].
For V = 0, the spectrum is the closed interval [−2, 2] ∈ R, and forV > 0, recall that the spectrum is a Cantor set.
To numerically approximate fractal measures for these sets, we usea MATLAB function crossSection to discretize the spectrum fora fixed V .
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Cross Sections of the Spectrum
The function crossSection iterates the trace map for initialpoints, x0 := E − V , y0 := E + V and outputs a representation ofthe spectrum.
We fix the value of V ∈ (0, 1) while varying E ∈ [−2, 2] to definedifferent initial points. Recall that if an initial point is notunstable, the corresponding E is in the spectrum.
After iterating the trace map, crossSection outputs a vector of0’s and 1’s, representing a cross section of the spectrum.
A value in the spectrum is represented by 1, while 0 denotes avalue not in the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Cross Sections of the Spectrum
The function crossSection iterates the trace map for initialpoints, x0 := E − V , y0 := E + V and outputs a representation ofthe spectrum.
We fix the value of V ∈ (0, 1) while varying E ∈ [−2, 2] to definedifferent initial points. Recall that if an initial point is notunstable, the corresponding E is in the spectrum.
After iterating the trace map, crossSection outputs a vector of0’s and 1’s, representing a cross section of the spectrum.
A value in the spectrum is represented by 1, while 0 denotes avalue not in the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Cross Sections of the Spectrum
The function crossSection iterates the trace map for initialpoints, x0 := E − V , y0 := E + V and outputs a representation ofthe spectrum.
We fix the value of V ∈ (0, 1) while varying E ∈ [−2, 2] to definedifferent initial points. Recall that if an initial point is notunstable, the corresponding E is in the spectrum.
After iterating the trace map, crossSection outputs a vector of0’s and 1’s, representing a cross section of the spectrum.
A value in the spectrum is represented by 1, while 0 denotes avalue not in the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
The Trace MapVisualizing the Spectrum
Cross Sections of the Spectrum
The function crossSection iterates the trace map for initialpoints, x0 := E − V , y0 := E + V and outputs a representation ofthe spectrum.
We fix the value of V ∈ (0, 1) while varying E ∈ [−2, 2] to definedifferent initial points. Recall that if an initial point is notunstable, the corresponding E is in the spectrum.
After iterating the trace map, crossSection outputs a vector of0’s and 1’s, representing a cross section of the spectrum.
A value in the spectrum is represented by 1, while 0 denotes avalue not in the spectrum.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Definition of Box-Counting Dimension
Definition (Box-Counting Dimension from [6])
The box-counting dimension of a Cantor set, K, is given by,
dimBK = limδ→0
logNδ(K)
− log δ,
if the limit exists, where Nδ(K) is the number of covers of lengthδ needed to cover the set.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
Using the function crossSection we obtain a vectorrepresentation of the spectrum:
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We then define a diameter δ and determine the minimum numberof δ-covers needed to cover the set.
In this discrete calculation, δ will correspond to a number ofentries from ~E to include in a cover.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
Using the function crossSection we obtain a vectorrepresentation of the spectrum:
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We then define a diameter δ and determine the minimum numberof δ-covers needed to cover the set.
In this discrete calculation, δ will correspond to a number ofentries from ~E to include in a cover.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
Using the function crossSection we obtain a vectorrepresentation of the spectrum:
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We then define a diameter δ and determine the minimum numberof δ-covers needed to cover the set.
In this discrete calculation, δ will correspond to a number ofentries from ~E to include in a cover.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
We construct ~E to have 2k entries. We now define δ as 2−i forlarger and larger i (thus taking δ to 0).
A δ-cover will include 2k × 2−i entries from ~E.
For example, if δ = 2−3 and ~E has 24 entries, each δ-coverincludes 2 entries.
~E =[ 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 ]
The minimum number of δ-covers is 4.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
For each δi, we determine Nδi(~E), the minimum number of
δi-covers needed to cover ~E.
We then plot the points (− log δi , logNδi(~E)) for each i and find
the slope of the regression line for these points.
This estimates the box-counting dimension, i.e.
limδ→0
logNδ(K)
− log δ
for the spectrum at the particular coupling constant we used togenerate ~E.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
For each δi, we determine Nδi(~E), the minimum number of
δi-covers needed to cover ~E.
We then plot the points (− log δi , logNδi(~E)) for each i and find
the slope of the regression line for these points.
This estimates the box-counting dimension, i.e.
limδ→0
logNδ(K)
− log δ
for the spectrum at the particular coupling constant we used togenerate ~E.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Box-Counting Dimension
For each δi, we determine Nδi(~E), the minimum number of
δi-covers needed to cover ~E.
We then plot the points (− log δi , logNδi(~E)) for each i and find
the slope of the regression line for these points.
This estimates the box-counting dimension, i.e.
limδ→0
logNδ(K)
− log δ
for the spectrum at the particular coupling constant we used togenerate ~E.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Results for Box-Counting Dimension
We find that as coupling constant approaches zero, box-countingdimension approaches one.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
0.85
0.9
0.95
1
Coupling Constant, V
Box
Cou
ntin
g D
imen
sion
Box Counting Dimension as Coupling Constant Approaches Zero
Student Version of MATLAB
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
ThicknessPreliminary Definitions
Let K ⊂ R be a Cantor set.
Definition (Bounded Gap from [9])
A bounded gap of K is a bounded connected component of R\K.
Definition (Bridge from [9])
Let U be any bounded gap of K and u be a boundary point of U ,that is u ∈ K. Then a bridge of K at u, denoted C, is themaximal interval in R such that
u is a boundary point of C, and
C contains no point of another gap U ′, whose length, `(U ′),is at least the length U (where length is defined in the usualway for R).
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
ThicknessPreliminary Definitions
Let K ⊂ R be a Cantor set.
Definition (Bounded Gap from [9])
A bounded gap of K is a bounded connected component of R\K.
Definition (Bridge from [9])
Let U be any bounded gap of K and u be a boundary point of U ,that is u ∈ K. Then a bridge of K at u, denoted C, is themaximal interval in R such that
u is a boundary point of C, and
C contains no point of another gap U ′, whose length, `(U ′),is at least the length U (where length is defined in the usualway for R).
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Definition of Thickness
Define thickness of K at a particular u to be
τ(K,u) =`(C)
`(U)
where `(C) is the length of the bridge and `(U) the length of itsrespective gap [9].
Definition (Thickness from [9])
The thickness of K, denoted τ(K) is defined as:
inf τ(K,u)
for all boundary points u of bounded gaps.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Definition of Thickness
Define thickness of K at a particular u to be
τ(K,u) =`(C)
`(U)
where `(C) is the length of the bridge and `(U) the length of itsrespective gap [9].
Definition (Thickness from [9])
The thickness of K, denoted τ(K) is defined as:
inf τ(K,u)
for all boundary points u of bounded gaps.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [
2 2 1 3 3
]
interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2
2 1 3 3
]
interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2
1 3 3
]
interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2 1
3 3
]
interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2 1 3 3 ]
interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To approximate the thickness of the spectrum, we first determinethe length of the gaps and intervals.
Using the function crossSection we obtain a representation ofthe spectrum,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
To simplify the process, first consider the thickness of rightboundary points.
For Example,
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
where we calculate,
gap = [ 2 2 1 3 3 ]interval = [ 1 2 1 1 ]
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We obtain the following:
gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]
To calculate the thickness, we divide the ith bridge length by the(i+ 1)th gap and take the minimum of these values.
For left boundary points, we reflect the sequence and repeat thesame process.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We obtain the following:
gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]
To calculate the thickness, we divide the ith bridge length by the(i+ 1)th gap and take the minimum of these values.
For left boundary points, we reflect the sequence and repeat thesame process.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Thickness
~E =[ 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 ]
We obtain the following:
gap = [ 2 2 1 3 3 ]bridge = [ 4 1 1 ]
To calculate the thickness, we divide the ith bridge length by the(i+ 1)th gap and take the minimum of these values.
For left boundary points, we reflect the sequence and repeat thesame process.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Resulting Trends
We find that as coupling constant approaches zero, thicknessapproaches infinity,
Thickness
V n = 1000 n = 10000
0.03 0.0106 0.0116
0.003 0.0833 0.0125
0.0003 3.0 0.0909
0.00003 383 1.0
Issue: Theoretically, the calculation of thickness should improvewith a better approximation of the Cantor Set.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Resulting Trends
We find that as coupling constant approaches zero, thicknessapproaches infinity,
Thickness
V n = 1000 n = 10000
0.03 0.0106 0.0116
0.003 0.0833 0.0125
0.0003 3.0 0.0909
0.00003 383 1.0
Issue: Theoretically, the calculation of thickness should improvewith a better approximation of the Cantor Set.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Introduction to Hausdorff Dimension
Consider covers of a set U 6= ∅ ⊂ R
Definition (Diameter from [6])
The diameter of U , denoted |U |, is defined as:
|U | := sup{|x− y| : x, y ∈ U}
Take diameters 0 ≤ |Ui| ≤ δ.
Definition (Hausdorff Measure from [6])
Let K ⊂ R be a set, and let s > 0. We define Hausdorff Measure,Hs(K), in the following way:
Hs(K) := limδ→0
(inf
{ ∞∑i=1
|Ui|s : {Ui} is a δ cover
})
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Introduction to Hausdorff Dimension
Consider covers of a set U 6= ∅ ⊂ R
Definition (Diameter from [6])
The diameter of U , denoted |U |, is defined as:
|U | := sup{|x− y| : x, y ∈ U}
Take diameters 0 ≤ |Ui| ≤ δ.
Definition (Hausdorff Measure from [6])
Let K ⊂ R be a set, and let s > 0. We define Hausdorff Measure,Hs(K), in the following way:
Hs(K) := limδ→0
(inf
{ ∞∑i=1
|Ui|s : {Ui} is a δ cover
})
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Hausdorff Dimension
Definition (Hausdorff Dimension from [6])
Let K ⊂ R, s > 0. Hausdorff dimension is given by
dimH(K) := inf {s ≥ 0 : Hs(K) = 0} = sup {s : Hs(K) =∞} .
It is known that the following relationship between Hausdorffmeasure and Hausdorff dimension holds:
Hs(K) =
{∞ if 0 ≤ s < dimH K0 if s > dimH K
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Hausdorff Dimension
Definition (Hausdorff Dimension from [6])
Let K ⊂ R, s > 0. Hausdorff dimension is given by
dimH(K) := inf {s ≥ 0 : Hs(K) = 0} = sup {s : Hs(K) =∞} .
It is known that the following relationship between Hausdorffmeasure and Hausdorff dimension holds:
Hs(K) =
{∞ if 0 ≤ s < dimH K0 if s > dimH K
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Graph of Hausdorff Dimension
∞
dimH(K)s
Hs(K)
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Box-Counting Dimension
In similar models (e.g. the Fibonacci Hamiltonian) thebox-counting and Hausdorff dimensions were found to coincide [5].
Recall, we found that as coupling constant approaches zero,box-counting dimension approaches one.
Therefore, we hypothesize that the Hausdorff dimension of ourspectrum will be approximately one.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Box-Counting Dimension
In similar models (e.g. the Fibonacci Hamiltonian) thebox-counting and Hausdorff dimensions were found to coincide [5].
Recall, we found that as coupling constant approaches zero,box-counting dimension approaches one.
Therefore, we hypothesize that the Hausdorff dimension of ourspectrum will be approximately one.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Box-Counting Dimension
In similar models (e.g. the Fibonacci Hamiltonian) thebox-counting and Hausdorff dimensions were found to coincide [5].
Recall, we found that as coupling constant approaches zero,box-counting dimension approaches one.
Therefore, we hypothesize that the Hausdorff dimension of ourspectrum will be approximately one.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Thickness
Theorem (Palis, Takens)
Thickness provides a lower bound of the Hausdorff dimension(dimH K) in the following way:
dimH K ≥log(2)
log(2 + 1τ(K))
According to our results,
limV→0
τ(K) =∞
it follows that when the coupling constant is zero,
limV→0
dimH(K) ≥ log(2)
log(2)= 1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Thickness
Theorem (Palis, Takens)
Thickness provides a lower bound of the Hausdorff dimension(dimH K) in the following way:
dimH K ≥log(2)
log(2 + 1τ(K))
According to our results,
limV→0
τ(K) =∞
it follows that when the coupling constant is zero,
limV→0
dimH(K) ≥ log(2)
log(2)= 1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Estimate of Hausdorff DimensionUsing Thickness
Theorem (Palis, Takens)
Thickness provides a lower bound of the Hausdorff dimension(dimH K) in the following way:
dimH K ≥log(2)
log(2 + 1τ(K))
According to our results,
limV→0
τ(K) =∞
it follows that when the coupling constant is zero,
limV→0
dimH(K) ≥ log(2)
log(2)= 1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Methods of Approximating Hausdorff Dimension
We numerically estimate the Hausdorff dimension of the spectrumusing methods employed by Chorin [3].
Consider the quantity nδs, where n is the number of covers neededfor diameter δ, and s is a nonnegative number.
We have the following bounds on the Hausdorff dimension
Lower bound: the value of s1, where nδs1 is strictly increasing
Upper bound: the value of s2 where nδs2 is strictly decreasing.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Methods of Approximating Hausdorff Dimension
We numerically estimate the Hausdorff dimension of the spectrumusing methods employed by Chorin [3].
Consider the quantity nδs, where n is the number of covers neededfor diameter δ, and s is a nonnegative number.
We have the following bounds on the Hausdorff dimension
Lower bound: the value of s1, where nδs1 is strictly increasing
Upper bound: the value of s2 where nδs2 is strictly decreasing.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Methods of Approximating Hausdorff Dimension
We numerically estimate the Hausdorff dimension of the spectrumusing methods employed by Chorin [3].
Consider the quantity nδs, where n is the number of covers neededfor diameter δ, and s is a nonnegative number.
We have the following bounds on the Hausdorff dimension
Lower bound: the value of s1, where nδs1 is strictly increasing
Upper bound: the value of s2 where nδs2 is strictly decreasing.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
Recall that HV = −∆ + V .
We approximate the spectrum by first truncating the Hamiltonianoperator:
Generate a finite string of the period doubling sequence.For example, the period doubling sequence after 3 iterations:
S3(1) =[ 1 -1 1 1 1 -1 1 -1 ]
Call the function truncatedHamiltonian which
Creates a matrix with S3(1) on the main diagonal
Multiples the matrix by V
Inputs 1’s along the super and sub-diagonals
Computes the eigenvalues of the resulting matrix.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
At this point, we employ a numerical approximation method [3].
We will be computing the quantity: nδs
We call the function hausdorffDimension which:
Defines δ = 2−i for i ∈ {1, 2, ..., 10}Calculates the number of covers, n, needed for each δ
Computes nδs, for s ∈ [0, 1].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
At this point, we employ a numerical approximation method [3].
We will be computing the quantity: nδs
We call the function hausdorffDimension which:
Defines δ = 2−i for i ∈ {1, 2, ..., 10}Calculates the number of covers, n, needed for each δ
Computes nδs, for s ∈ [0, 1].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
At this point, we employ a numerical approximation method [3].
We will be computing the quantity: nδs
We call the function hausdorffDimension which:
Defines δ = 2−i for i ∈ {1, 2, ..., 10}
Calculates the number of covers, n, needed for each δ
Computes nδs, for s ∈ [0, 1].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
At this point, we employ a numerical approximation method [3].
We will be computing the quantity: nδs
We call the function hausdorffDimension which:
Defines δ = 2−i for i ∈ {1, 2, ..., 10}Calculates the number of covers, n, needed for each δ
Computes nδs, for s ∈ [0, 1].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Using MATLAB to Approximate Hausdorff Dimension
At this point, we employ a numerical approximation method [3].
We will be computing the quantity: nδs
We call the function hausdorffDimension which:
Defines δ = 2−i for i ∈ {1, 2, ..., 10}Calculates the number of covers, n, needed for each δ
Computes nδs, for s ∈ [0, 1].
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Results for Hausdorff Dimension
The value of the quantity: nδs where δ = 2−i, is displayed in thefollowing table (V = 0.001).
si 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
1 4.757 4.724 4.691 4.659 4.627 4.595 4.563 4.531 4.5002 4.748 4.683 4.619 4.555 4.492 4.430 4.369 4.309 4.2503 4.872 4.771 4.673 4.577 4.483 4.391 4.300 4.212 4.1254 5.071 4.933 4.798 4.667 4.539 4.415 4.294 4.177 4.0635 5.319 5.138 4.963 4.794 4.631 4.473 4.321 4.173 4.0316 5.601 5.373 5.154 4.944 4.742 4.549 4.364 4.186 4.0167 5.909 5.629 5.362 5.108 4.866 4.636 4.416 4.207 4.008
Therefore, the Hausdorff dimension is bounded below by 0.92 andabove by 1.00.
As V approaches 0, Hausdorff dimension approaches 1.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Results for Hausdorff Dimension
The value of the quantity: nδs where δ = 2−i, is displayed in thefollowing table (V = 0.001).
si 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
1 4.757 4.724 4.691 4.659 4.627 4.595 4.563 4.531 4.5002 4.748 4.683 4.619 4.555 4.492 4.430 4.369 4.309 4.2503 4.872 4.771 4.673 4.577 4.483 4.391 4.300 4.212 4.1254 5.071 4.933 4.798 4.667 4.539 4.415 4.294 4.177 4.0635 5.319 5.138 4.963 4.794 4.631 4.473 4.321 4.173 4.0316 5.601 5.373 5.154 4.944 4.742 4.549 4.364 4.186 4.0167 5.909 5.629 5.362 5.108 4.866 4.636 4.416 4.207 4.008
Therefore, the Hausdorff dimension is bounded below by 0.92 andabove by 1.00.
As V approaches 0, Hausdorff dimension approaches 1.Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Future Work
Further work for this problem includes refining the approximationsof the two fractal dimensions, refining the approach to thethickness computations, and analyzing the physical implications ofthe results.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Acknowledgements
We would like to thank our advisor, May Mei, and our project TA,Drew Zemke. Thanks also to Ravi Ramakrishna, Summer MathInstitute program director, and the math department at CornellUniversity. This work was supported by NSF grant DMS-0739338.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Michael Baake, Uwe Grimm, and Robert V. Moody.What is aperiodic order?Spektrum der Wissenschaft, pages 64–74, 2002.The translated version was used for our research.
Jean Bellissard, Anton Bovier, and Jean-Michel Ghez.Spectral properties of a tight binding hamiltonian with perioddoubling potential.Communications in Mathematical Physics, 135(2):379–399,1991.
Alexandre Joel Chorin.Numerical estimates of hausdorff dimension.Journal of Computational Physics, 46(3):390 – 396, 1982.
Jean-Michel Combes.Connections between quantum dynamics and spectralproperties of time-evolution operators.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
In E.M. Harrell W.F. Ames and J.V. Herod, editors,Differential Equations with Applications to MathematicalPhysics, volume 192 of Mathematics in Science andEngineering, pages 59 – 68. Elsevier, 1993.
Daminik, Embree, and Gorodetski.Spectral properties of schrodinger operators arising in thestudy of quasicrystalsl.1210.5753, October 2012.
Kenneth Falconer.Hausdorff Measure and Dimension, pages 27–38.John Wiley & Sons, Ltd, 2005.
I. Guarneri.Spectral properties of quantum diffusion on discrete lattices.EPL, 10(2):95–100, 1989.
Yoram Last.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian
Period Doubling HamiltonianThe Spectrum
Methods and Results
Box-Counting DimensionThicknessHausdorff Dimension
Quantum dynamics and decompositions of singular continuousspectra.Journal of Functional Analysis, 142(2):406 – 445, 1996.
Jacob Palis and Floris Takens.Hyperbolicity and sensitive chaotic dynamics at homoclinicbifurcations, volume 35 of Cambridge Studies in AdvancedMathematics.Cambridge University Press, Cambridge, 1993.Fractal dimensions and infinitely many attractors.
Within ε of Awesome Spectrum of the Period Doubling Hamiltonian