The spiral galaxy blowup of SG Andrew Hahm

Jeffrey Kuan

August 10, 2016

Andrew Hahm, Jeffrey Kuan

The spiral galaxy blowup of SG

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Outline

1

Motivation and background

2

The Laplacian on SG∞

3

Heat kernel on SG∞

4

Schr¨odinger operators on SG∞

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Motivation

1

Spectrum of the Laplacian on the interval [0, 1]

2

Fourier series:

∞ X

f (x) ∼

fˆ(n)e 2πinx

(1)

fˆ(ξ)e 2πixξ dξ

(2)

n=−∞ 3

Spectrum of the Laplacian on R

4

Fourier transform: Z

∞

f (x) ∼ −∞

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Blowups of the Sierpi´nski gasket

Let q0 , q1 , q2 ∈ R2 be vertices of SG, and let Fi , i = {0, 1, 2} be the contractive mappings defining SG such that 1 Fi : x 7→ (x − qi ) + qi . 2

(3)

Given an infinite sequence {w1 , . . . , wn , . . .}, wi ∈ {0, 1, 2}, we define a blowup of SG as the infinite union ∞ [

Fw−1 ◦ · · · ◦ Fw−1 (SG ). 1 i

(4)

i=1

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Spiral galaxy blowup SG∞

If the sequence w = {w1 , . . . , wn , . . .} is constructed by repeating a finite word infinitely many times, it defines a periodic blowup. We define SG∞ as the periodic blowup ∞ [

(F0−1 ◦ F1−1 ◦ F2−1 )m (SG ).

(5)

m=1

We also define the level-k blowup approximations SGk as k [

(F0−1 ◦ F1−1 ◦ F2−1 )m (SG ).

(6)

m=1

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Energy Recall that we defined the m-th energy approximation of a function on SG as the quadratic form X Em (u) = (u(x) − u(y ))2 , (7) x∼m y

where ∼m ranges over edges of the m-th graph approximation of SG. Then we defined the renormalized energies on SG as Em (u) = r −m Em (u)

(8)

for r = 53 . Also recall that we defined h to be a harmonic function on SG if Em = En for all m 6= n.

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Energy measures

It is straightforward to check that, given a function u on SG, νu defined so that νu (SG ) = E(u) νu (Fw SG ) = r

−|w |

(9) E(u ◦ Fw )

(10)

is a measure. We call such a measure an energy measure. We want to work with Laplacians with respect to energy measures associated to harmonic functions.

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The SG∞ energy measure

Recall that each contraction mapping Fi can be represented as a matrix Ai , e.g.:   1 0 0 (11) A0 =  52 25 51  . 2 5

1 5

2 5

Each Ai is invertible and a linear transformation on the space of harmonic functions. Thus the zoom-out map (F2 ◦ F1 ◦ F0 )−1 for one SG∞ blowup of SG , has a matrix form (A2 · A1 · A0 )−1 . This matrix, after an appropriate change of basis, has two nonconstant harmonic eigenfunctions, h1 , h2 . We define ν = νh1 + νh2 as an energy measure on SG1 .

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The SG∞ energy measure on SG1

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Approximations of SG∞

How are we computing our Laplacian? There are three approximations we make: 1

Finite blowup approximations via pointwise formula: Z −1 (m) −m ∆ν f (x) = lim r ψx dν ∆m f (x). m→∞

(12)

2

Numerical integration of spline function in pointwise formula

3

Dirichlet boundary conditions for rapidly decreasing eigenfunctions

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Eigenvalues of graph Laplacians on SG1

level 8 0.03722615167 0.1162320545 0.1433856317 0.2699408738 0.3432478029 0.4891472988 0.5050212546 0.6707739058 0.7707858347 0.9485626201 1.096346518 1.153654906

level 7 0.03722243382 0.1162013025 0.1433677887 0.2697948534 0.3431520604 0.4887557521 0.5048137997 0.6703050852 0.7700868353 0.947925638 1.094400189 1.151700347

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level 6 0.03720837389 0.1160852992 0.1433003653 0.269248097 0.3427897426 0.4872912498 0.5040409271 0.6685668361 0.7674978133 0.9455898424 1.087141667 1.144776681

level 5 0.03715543846 0.1156525379 0.1430473813 0.2672663081 0.3414280479 0.4820410472 0.5012749785 0.6625050303 0.7586068339 0.9377046456 1.062753162 1.123029027

The spiral galaxy blowup of SG

level 4 0.03695932254 0.114105684 0.1421205535 0.260930505 0.3365227907 0.4661690076 0.4923642583 0.6444060613 0.7342012585 0.9142314582 1.008087525 1.053645572

level 3 0.0362741543 0.109417813 0.1388577187 0.2474993549 0.3198350337 0.4273827536 0.4638338055 0.5660961252 0.6535589182 0.7224977201 0.808947565 0.9540018202

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Ground state eigenfunction, level 3

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Ground state eigenfunction, level 4

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Ground state eigenfunction, level 5

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Ground state eigenfunction, level 6

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Ground state eigenfunction, level 7

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Miniaturization

We expect that these eigenfunctions miniaturize with c=

56 . 33

(13)

This is due to the pointwise approximation we are using: Z −1 (m) −m r ψx dν ∆m f (x).

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(14)

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Examples of miniaturization: level 6, λ = 37.84826

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Examples of miniaturization: level 9, λ = 21903.12

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Examples of miniaturization: level 3, λ = 1.676421

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Examples of miniaturization: level 6, λ = 970.3086

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Examples of miniaturization: level 9, λ = 561521.2

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Peano curves

We want a better way to compare miniaturization pairs. Peano curves give us one way to do so. A Peano curve of SG∞ is a continuous, surjective mapping γ : [0, ∞) → SG∞ .

(15)

[Molinor, Ott, Strichartz 2014] provides a construction of a Peano curve γ : S 1 → SG via Peano curves on approximations.

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Peano curve schematic

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Peano curve approximation, level 7

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Peano curves of miniaturization pairs: levels 6 and 9

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Examples of miniaturization: level 6, λ = 37.84826

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Examples of miniaturization: level 9, λ = 21903.12

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Peano curves of miniaturization pairs: levels 3 and 6

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Examples of miniaturization: level 3, λ = 1.676421

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Examples of miniaturization: level 6, λ = 970.3086

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Peano curves of miniaturization pairs: levels 6 and 9

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Examples of miniaturization: level 6, λ = 970.3086

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Examples of miniaturization: level 9, λ = 561521.2

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Heat kernel The heat kernel Ht (x, y ) =

X

e −λj t uj (x)uj (y )

(16)

j

can be used to solve the heat equation ∂u(x, t) = c∆u(x, t) ∂t

(17)

via integration against the kernel: Z u(x, t) = Ht (x, y )f (y )dν(y ),

(18)

for initial value condition u(x, 0) = f (x).

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Heat kernel, power law We want to see if there exists a power law Ht (x, x) ≈ t −α .

(19)

We experimentally found that, for 0 < t < 0.4, our α-values fall in the range 0.50 < α < 1.26. (20)

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Heat kernel, power law

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Heat kernel, power law

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Heat kernel, off-diagonal behavior

Link

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Heat kernel, off-diagonal behavior, log scale

Link

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Schr¨odinger operators We define the Schr¨odinger operators −∆ν + cV ,

(21)

where c > 0 is an arbitrary constant, and V is the potential V (x) = h1 (x)2 + h2 (x)2 .

(22)

Notice that 1 2

V (x) ≥ 0, and given a sequence of points {xm }∞ m=1 such that xm ∈ SGm \SGm−1 , V (xm ) → ∞ as m → ∞.

[Fan, Khandker, Strichartz 2009] discusses the construction of similar harmonic-oscillator Hamiltonians on infinite blowups with the standard measure. Andrew Hahm, Jeffrey Kuan

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Schr¨odinger operators, c=0.5, ground state e.f.

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Schr¨odinger operators, c=1, ground state e.f.

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Schr¨odinger operators, c=2, ground state e.f.

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Schr¨odinger operators, c=10, ground state e.f.

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Schr¨odinger operators, c=0.5, ground state decay

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Schr¨odinger operators, c=1, ground state decay

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Schr¨odinger operators, c=2, ground state decay

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Schr¨odinger operators, c=10, ground state decay

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Schr¨odinger operators, eigenvalue counting function

Let {λj } be an enumeration of eigenvalues with the usual ordering ≤. We define the eigenvalue counting function ρ(x) as ρ(x) = #{λj : λj ≤ x}.

(23)

We want to calculate an analog to the Weyl ratio in smooth analysis, so we seek a value β such that ρ(x)/x β is periodic relative to log(x).

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Schr¨odinger operators, Weyl ratio Recall the miniaturization constant c=

56 . 33

(24)

Thus, we expect the “eigenvalue counting function” for the potential V to follow: ρ(ct) ≈ 27ρ(t) ⇒ 27ρ(t) ρ(ct) ≈ β β ⇒ β β c t c t ρ(ct) 27 1= ≈ β. ρ(t) c This gives β ≈

log(27) log(c)

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≈ 0.51815.

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Schr¨odinger operators, potential only β = 0.51462465095

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Schr¨odinger operators, c=10 β = 0.519424536066

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Schr¨odinger operators, c=2 β = 0.515630567958

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Schr¨odinger operators, c=1 β = 0.519970636742

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Future avenues of research

1

Understand and prove quantitative statements about the numerics of the SG∞ energy measure

2

Prove quantitative statements or bounds on limiting behavior of Laplacian, its spectrum, its eigenfunctions, and the heat kernel

3

Construct and explore different Sch¨ odinger operators

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