Asymptotic Spectral Properties of the Schrödinger Operator with Thue-Morse Potential William Clark Rachael Kline Michaela Stone Cornell SMI Advisors: May Mei and Andrew Zemke August 2, 2013
Introduction History 1984 An AIMn alloy demonstrated aperiodicity under x-ray diffraction, giving rise to new questions about material structure. This generated interest in mathematical exploration of aperiodic tilings. Figure : Al 6 Mn.
Introduction What is aperiodicity? Figure : Diffraction pattern of a periodic crystal.
Introduction What is aperiodicity? Figure : Diffraction pattern of an aperiodic quasicrystal.
Introduction Why do we care? Understanding the mathematics behind aperiodic tilings allows us to model the behavior of electrons traveling through quasicrystals. Properties: (Marder 2011) Directionally varying electrical conductivity Poor heat conductivity
Introduction Why do we care? Understanding the mathematics behind aperiodic tilings allows us to model the behavior of electrons traveling through quasicrystals. Properties: (Marder 2011) Uses: Directionally varying electrical conductivity Poor heat conductivity Surgical instruments LED lights Non-stick cookware
Introduction How do we mathematically model quasicrystals? 2-dimensional modeling: Tilings of the plane 1-dimensional modeling: Sequences
Introduction Constructing sequences A possible construction of these structures is generated using inflation rules that utilize a finite alphabet to represent the tiling. A substitution, σ, assigns a finite string of letters to each letter in the alphabet. Definition Letter - a single representative of the alphabet (a, b, c,...) Word - a collection of letters (ab, ba, abc,...)
Introduction One specific sequence, the Thue-Morse sequence, uses two letters, a and b with the substitutions σ(a) = ab, σ(b) = ba. We denote the sequence as u where u n is the n th letter in the sequence. Suppose u 0 = a, using the substitution we obtain the sequence: u = abbabaabbaababba... where the sequence is the limit of the substitutions as the length of the sequence goes to infinity.
Developing the Trace Map The 1D discrete Schrödinger operator with a Thue-Morse potential defined on the discrete 1-D lattice Z where V is the coupling constant, is H V ψ(n) = ψ(n + 1) + V (n)ψ(n) + ψ(n 1) where V (n) = V ( n 1) and V (n) = V whenever the n th letter of u is a and V (n) = V otherwise (by convention u 0 = a) [2].
Developing the Trace Map The Schrödinger equation in terms of transfer matrices Definition Let T (n) be the transfer matrix: [ ] E V (n) 1. 1 0 Then we can write the Schrödinger equation in terms of T (n) as follows: [ ] [ ] [ ] [ ] ψ(n + 1) E V (n) 1 ψ(n) ψ(n) = = T (n). ψ(n) 1 0 ψ(n 1) ψ(n 1)
Developing the Trace Map From matrix to map Using our understanding of the recursion in the Thue-Morse sequence and through application of trace properties and matrix identities, we are able to develop a trace map from R 2 to itself defined by: f (x, v) = (x 2 2 v, v(v + 4 x 2 )) with the initial conditions v 1 = 4V 2, and x 1 = E 2 V 2 2, where E is the energy and V is a coupling constant, a real parameter.
Developing the Trace Map Definition A point (x, v) is called unstable if there is a neighborhood U of (x, v) and an integer n 0 such that for all (x 1, v 1 ) in U and all N n 0, the iterated (x N, v N ) = f N (x 1, v 1 ) satisfies x N > 2. Proposition E, energy, values are out of the spectrum if and only if they correspond to an unstable (x, v) point. (Bellissard 1990)
Developing the Trace Map D 0 +,+ +,- +,- -,+ -,- -,-
Developing the Trace Map
Developing the Trace Map
Matrix Truncation Truncation The matrix representation of the Schrödinger operator acting on the Thue-Morse Sequence is the tridiagonal n n matrix:........................ V(-2) 1 0 0 0...... 1 V(-1) 1 0 0...... 0 1 V(0) 1 0...... 0 0 1 V(1) 1...... 0 0 0 1 V(2)........................ The spectrum of a matrix is defined as the set of its eigenvalues. By analysing the spectrum of these finite n n matrices, as n goes to infinity, we can approximate the spectrum of the operator.
Fractal Dimension Fractal Dimension Since the spectrum is a Cantor set with measure zero [2], we wish to analyse the fractal dimension of the spectrum for different coupling constants, V. The notions of fractal dimension we will be using include: Box-counting dimension Thickness Hausdorff dimension
Fractal Dimension How dimensions are related In an arbitrary set, F, we have the following relation: log 2 ( log 2 + 1 ) dim H (F ) dim B (F ) : τ where τ is the thickness, dim H (F ) is the Hausdorff dimension, dim B (F ) is box-counting dimension (Falconer 2005, Palis et al. 1993).
Fractal Dimension Box-counting Dimension The box-counting dimension of a set F is: dim B F = lim δ 0 log N δ (F ) log δ, where N δ is the minimum cardinality of a δ-cover of F. This equation is assuming that the measure of the set F obeys the power law: N δ (F ) cδ s log(n δ ) log(c) s log(δ)
Fractal Dimension
Fractal Dimension Box-counting Dimension
Fractal Dimension Box-counting dimension
Fractal Dimension Thickness Another type of fractal dimension we will consider is the thickness of a Cantor set. Definition Let K R. A gap of K is a connected component R\K; a bounded gap is a bounded connected component of R\K. Let U be any bounded gap and u be a boundary point of U, so u K. Let C be the bridge of K at u, i.e. the maximal interval in R such that - u is a boundary point of C, - C contains no point of a gap U whose length l(u ) is at least the length of U [4].
Fractal Dimension Thickness Definition The thickness of K at u is defined as τ(k, u) = l(c)/l(u). The thickness of K, denoted by τ(k), is the infimum over these τ(k, u) for all boundary points u of bounded gaps.
Fractal Dimension Thickness Definition The thickness of K at u is defined as τ(k, u) = l(c)/l(u). The thickness of K, denoted by τ(k), is the infimum over these τ(k, u) for all boundary points u of bounded gaps. We cannot compute this directly as a lower bound for the Hausdorff dimension. We plug the calculation of thickness into the function: log 2 ( log 2 + 1 ) τ
Fractal Dimension Thickness
Fractal Dimension Hausdorff Dimension Before the Hausdorff dimension can be defined, the Hausdorff measure of a set must be defined [3]: Definition The Hausdorff measure of a set F of dimension s, denoted by H s (F ), is: { { }} H s (F ) = lim inf U i s : {U i } is a δ-cover of F δ 0 i=1
Fractal Dimension Hausdorff Dimension With this in mind, we can define the Hausdorff dimension as follows[3]: Definition The Hausdorff dimension of a set F, denoted by dim H (F ), is: dim H (F ) = inf{s 0 : H s (F ) = 0} = sup{s : H s (F ) = } Specifically, dim H (F ) is the value of s where the Hausdorff measure jumps from to 0.
Fractal Dimension Using the Trace Map to compute Hausdorff dimension Suppose V = 0.03: s-values i 0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 9 533 267.752 134.560 67.654 34.031 17.127 8.625 4.346 2.191 1.106 10 533 247.905 115.351 53.697 25.008 11.653 5.433 2.535 1.183 0.553 11 533 229.529 98.884 42.619 18.378 7.929 3.423 1.478 0.639 0.276 12 538 214.400 85.468 34.082 13.596 5.426 2.166 0.865 0.346 0.138 13 566 207.973 76.418 28.079 10.318 3.791 1.393 0.512 0.188 0.069 14 1132 356.567 112.314 35.378 11.144 3.510 1.106 0.348 0.110 0.035 15 2264 611.329 165.072 44.573 12.036 3.250 0.878 0.237 0.064 0.017 16 4528 1048.12 242.612 56.159 12.999 3.009 0.697 0.161 0.037 0.009 17 9056 1796.98 356.575 70.755 14.040 2.786 0.553 0.110 0.022 0.004
Fractal Dimension Using the Matrix Truncation to compute Hausdorff dimension Suppose V = 0.03, using a 1000 1000 matrix: s-values i 0.00 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 1 10 9.259 8.572 7.937 7.349 6.804 6.300 5.833 5.400 5.000 2 18 15.430 13.228 11.339 9.721 8.333 7.143 6.124 5.249 4.500 3 34 26.986 21.419 17.000 13.493 10.709 8.500 6.746 5.355 4.250 4 66 48.501 35.642 26.192 19.248 14.145 10.394 7.638 5.613 4.125 5 130 88.451 60.182 40.947 27.860 18.956 12.898 8.775 5.971 4.063 6 258 162.530 102.387 64.500 40.632 25.597 16.125 10.158 6.399 4.031 7 506 295.132 172.140 100.403 58.562 34.157 19.923 11.620 6.778 3.953 8 769 415.283 224.265 121.110 65.403 35.320 19.074 10.300 5.562 3.004 9 891 445.500 222.750 111.375 55.688 27.844 13.922 6.961 3.480 1.740
References What are Quasicrystals, and What Makes them Nobel-Worthy? J. Bellissard, Spectral Properties of Schrödinger s Operator with a Thue-Morse Potential, Number Theory and Physics (Jean-Marc Luck, Pierre Moussa, and Michel Waldschmidt, eds.), Springer Proceedings in Physics, vol. 47, Springer Berlin Heidelberg, 1990, pp. 140 150 (English). Keith Falconer, Fractal geometry, John Wiley & Sons, Ltd, 2005. Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993, Fractal dimensions and infinitely many attractors. MR 1237641 (94h:58129)
Acknowledgments We would like thank: Cornell University Cornell Summer Mathematics Institute May Mei Drew Zemke This work was supported by NSF grant DMS-0739338.