cut-and-project sets:

Transcription

1 cut-and-project sets: diffraction and harmonic analysis Christoph Richard, FAU Erlangen-Nürnberg Seminaire Lotharingien de Combinatoire, March / 30

2 motivation harmonic analysis? harmonic analysis of LCA groups heavy machinery, but quick proofs of fundamental results (density formula, pure point diffraction of regular model sets) standard tool for cut-and-project sets and Meyer sets 2 / 30

3 motivation diffraction experiments L A S E R Box 1 Experimental setup for optical diffraction The laser beam is widened by an arrangement of lenses and orthogonally laser or X -ray beam hits specimen (green) illuminates the object located at the green plane. The light that emanates fromatoms the object emitplane diffraction then interferes, waves (red) and the diffraction pattern is given by the waves distribution interfer of and light produce that one diffraction would observe picture at (purple) an infinite distance from the object. By another lens, this pattern is mapped onto the pink plane. Whereas for a picture of the object, as for instance in a camera, light rays emanating from one point of the object ideally are focused again 3 / 30

4 motivation Fraunhofer diffraction optics: Kirchhoff s approximation atom in x emits diffraction wave, modelled by waves interfer additively (structure factor) e 2π9ı k x observed intensity on screen at position k is absolute square 4 / 30

5 motivation diffraction of the Fibonacci chain Diffraktion für N=10 Intensität 0 1/ Wellenzahl 5 / 30

6 motivation diffraction of the Fibonacci chain Diffraktion für N=25 Intensität 0 1/ Wellenzahl 5 / 30

7 motivation diffraction of the Fibonacci chain Diffraktion für N=100 Intensität 0 1/ Wellenzahl 5 / 30

8 motivation diffraction of the Fibonacci chain Diffraktion für N=100 Intensität 0 1/ Wellenzahl pure point diffractive? 5 / 30

9 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (0,1) Wellenzahl position of peak p0, 1q p / 30

10 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (1,0) (0,1) Wellenzahl positions of peaks: p0, 1q{p1, 0q p : τ 6 / 30

11 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (1,0) (0,1) (1,1) Wellenzahl peak positions: pm, nq p 1 c pm ` nτq where c / 30

12 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (1,0) (0,1) (1,1) (1,2) Wellenzahl peak positions: pm, nq p 1 c pm ` nτq where c / 30

13 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (1,0) (0,1) (1,1) (2,3) (1,2) Wellenzahl peak positions: pm, nq p 1 c pm ` nτq where c / 30

14 motivation Fibonacci chain diffraction: indexing Bragg peaks Diffraktion für N=100 Intensität 0 1/2 1 (1,0) (0,1) (1,1) (2,3) (1,3) (1,2) Wellenzahl peak positions: pm, nq p 1 c pm ` nτq where c / 30

15 motivation mathematical diffraction theory Let Λ uniformly discrete such that even ΛΛ 1 is uniformly discrete ω ř ppλ δ p Dirac comb of Λ infer diffraction of Λ from finite samples ω n ω Bn convolution theorem yields Wiener diagram F ω n đ ÝÝÝÝÑ ω n Ăω n đf xω n 2 ÝÝÝÝÑ xω n xω n rωpf q ωp r f q with r f pxq f px 1 q identify Bragg peaks and continous components from the Lebesgue decomposition of the limiting measure of xω n xω n Fourier analysis of unbounded measures! 7 / 30

16 motivation mathematical diffraction theory F ω n đ ÝÝÝÝÑ ω n Ăω n đf xω n 2 ÝÝÝÝÑ xω n xω n Assume that the autocorrelation γ of ω exists as a vague limit 1 γ lim nñ8 θpb n q ω n Ăω n Since γ is positive definite, it is transformable, and by continuity of the Fourier transform we have ˆ ˆ 1 1 F lim nñ8 θpb n q ω n Ăω n lim F nñ8 θpb n q ω 1 n Ăω n lim nñ8 θpb n q xω n xω n We work with the autocorrelation as pω may not be a measure. This is in contrast to the case Λ a lattice. 8 / 30

17 Fourier analysis Fourier analysis on LCA groups: setup σ compact LCA group G with Haar measure θ G inverse function: r f pxq f px 1 q convolution: for f, g P L 1 pgq define f g P L 1 pgq by f gpxq f pyqgpy 1 xq dθ G pyq character χ : G Ñ Up1q continuous group homomorphism 9 / 30

18 Fourier analysis Pontryagin dual p G and Fourier transform pg set of all characters with topology induced by NpK, εq tχ P p P K : χpkq 1 ă εu for non-empty compact K Ă G and ε ą 0 pg LCA group with Haar measure θ pg Fourier transforms p f, q f : G p Ñ C of f P L 1 pgq p f pχq f pxqχpxq dθ G pxq, q f pχq f pxqχpxq dθ G pxq G Normalise θ pg such that Plancherel s formula f 2 p f 2 is satisfied for all f P L 1 pgq X L 2 pgq. G 10 / 30

19 Fourier analysis Fourier analysis of unbounded measures MpGq set of Borel measures on G Definition (cf. Argabright de Lamadrid 74) µ P MpGq is transformable if there exists pµ P Mp p Gq such that for all f P C c pgq such that q f P L 1 p p Gq we have q f P L 1 ppµq, xµ, f y xpµ, q f y. Poisson summation formula pµ uniquely determined by µ pµ translation bounded, i.e., for every compact K Ă G supt µ ptkq t P Gu ă 8, with µ P MpGq the total variation measure of µ 11 / 30

20 Fourier analysis examples of transformable measures µ positive definite, i.e., for all f P C c pgq f rf pxq dµpxq ě 0 G for example: δ Λ Dirac comb of a lattice Λ Ă G pδ Λ denspλq δ Λ 0 with dual lattice Λ 0 tχ P G p χppq 1@p P Λu classical examples finite measures, positive definite fctns, L p -fctns for p P r1, 2s 12 / 30

21 Fourier analysis The function space KLpG q KLpGq : tf P C c pgq p f P L 1 p p Gqu Such functions are not rare: If f, g P L 2 pgq have compact support, then f g P KLpGq. example: 1 W Ă1 W for relatively cpct measurable W Ă G. In fact KLpGq is dense in C c pgq. 13 / 30

22 Transformability and averages character averages Lemma Let χ P G. p Then for every van Hove sequence pa n q npn in G 1 lim χpxq dθpxq δ χ,e. nñ8 θpa n q A n This is obvious for χ e. Consider χ e for the following proof. 14 / 30

23 Transformability and averages By left invariance of the Haar measure and χpxyq χpxqχpyq χpyq dθpyq 1 An pxyqχpxyq dθpyq χpxq χpyq dθpyq A n G x 1 A n Due to the van Hove property of pa n q npn, we have ˇ χpyq dθpyq χpyq dθpyq ˇ ď θppx 1 A n q A n q opθpa n qq x 1 A n A n Combining the above properties yields 1 χpxq 1 ˇ χpyq dθpyq θpa n q ˇ op1q A n Lemma follows with x P G such that χpxq / 30

24 Transformability and averages The discrete part of pµ can be computed by averaging over µ: Proposition Let µ P MpGq be transformable and translation bounded and consider χ P G. p Then for every van Hove sequence pa n q npn in G we have 1 pµptχuq lim χpxqdµpxq nñ8 θpa n q A n history Argabright de Lamadrid 90 for pµ transformable Hof 95 for euclidean G Lenz 09 for µ the autocorrelation measure 16 / 30

25 Transformability and averages w.l.o.g. for χ e since pµptχuq pδ χ 1 pµqpteuq xχµpteuq smoothing of characteristic functions: f n 1 θpa n q 1 A n, pf n q ϕ ϕ f n, where ϕ ψ rψ with ψ P C c pgq and ş ψ 1. Then pf n q ϕ P KLpGq by the above lemma, and PSF yields µppf n q ϕ q pµ ~pfn q ϕ Consider the limit n Ñ 8 on the rhs: Since ˇ ~pf n q ϕ pχq Ñ δ χ,e, ˇ~pf n q ϕˇˇˇ qϕ ˇˇˇq fnˇˇˇ ď qϕ, we can use dominated convergence to infer lim ~pfn pµ q nñ8 ϕ pµpδ χ,e q pµpteuq Indeed by the previous lemma for χ e we have ˇ ˇ ˇ ˇ~pf n q ϕ pχqˇ qϕpχq ˇqf n pχqˇ ď ϕ 1 ˇqf n pχqˇ Ñ 0 17 / 30

26 Transformability and averages Consider the limit n Ñ 8 on the lhs of µppf n q ϕ q pµ ~pfn q ϕ f n and pf n q ϕ differ only near the boundary of A n, i.e., where K supppϕq Hence by a standard estimate f n pxq pf n q ϕ pxq ùñ x P B K A n µpf n q µppf n q ϕ q ď 1 ϕ 8 µ pbk A n q θpa n q rhs vanishes by translation boundedness of µ and by the van Hove property of pa n q npn. 18 / 30

27 model sets reminder: model sets assumptions: G, H σ-cpct LCA groups, H metrisable cut-and-project scheme with star map pq : L Ñ L π G π H G ÐÝ G ˆ H ÝÑ H Y Y Y L 1 1 ÐÝ lattice L dense ÝÑ L projection set via window W Ă H NpW q tx P L x P W u regular model set: W relatively cpct measurable, volpbw q 0 ( W ) We normalise Haar measure of H such that densplq / 30

28 model sets dual cut-and-project schemes duality theory for LCA groups leads to dual cut-and-project scheme Theorem (dual cut-and-project scheme) Let pg, H, Lq be a cut-and-project scheme and let L 0 P p G ˆ ph be the lattice dual to L. Then p p G, p H, L 0 q is also a cut-and-project scheme. diffraction is described within the dual cut-and-project scheme for euclidean groups p G» G and p H» H 20 / 30

29 model sets weighted model sets and transformability for cp scheme pg, H, Lq and h : H Ñ C define weighted model set ω h ÿ xpl hpx qδ x Theorem (R-Strungaru) Let pg, H, Lq be a cut-and-project scheme and let h P KLpHq. Then ω h is a transformable measure with xω h ω qh Here ω qh is the weighted model set of the dual cut-and-project scheme p p G, p H, L 0 q with weight function q h P L 1 p p Hq. 21 / 30

30 model sets proof of transformability for arbitrary g P KLpGq we have xω h, gy xδ L, g hy xδ L 0, qg qhy xω qh, qgy first equation: π G L one-to-one second equation: PSF and g h P KLpG ˆ Hq. third equation: π pg one-to-one ˇˇL0 equations also imply ω h P MpGq and qg P L 1 pω qh q hence xω h ω qh by definition 22 / 30

31 density formula weighted model sets and density formula Theorem (density formula) Let h P C c phq. Then for every van Hove sequence pa n q npn ω h pa n q lim nñ8 θ G pa n q hpxqdθ H pxq. H history Meyer 70 s for euclidean G, H via PSF (see also Matei Meyer 10, Lev Orlevskii 13) Schlottmann 98, geometric proof Moody 02 via dynamical systems Lenz R 07 for admissible h P L 1 bc phq via dynamical systems 23 / 30

32 density formula weighted model sets and density formula an immediate consequence: Corollary The density formula also holds for h 1 W where W Ă H is relatively cpct measurable with almost no boundary θ H pbw q 0. Consider arbitrary ε ą 0. Since h is Riemann integrable, we find ϕ, ψ P C c phq such that ϕ ď h ď ψ, pψpxq ϕpxqqdθ H pxq ď ε{2 H The density formula yields for sufficiently large n the estimate n q ε ď ε{2` ψ ωψpa θ G pa n q ď h ωhpa n q θ G pa n q ď ε{2` n q ϕ ωϕpa θ G pa n q ď ε 24 / 30

33 density formula proof for h P C c phq first step: proof for h P KLpHq by PSF Assume that h P KLpHq. Then pω h ω qh and hpxqdθ H pxq q ω h pa n q hpeq ω qh pteuq xω h pteuq lim nñ8 θpa n q H second step: extension to C c phq by approximation Use that KLpHq is dense in C c phq. consider the uniformly discrete Λ supppω h q Ď G and note denspλq lim sup nñ8 1 θ G pa n q Λ X A n ă 8 Take h P C c phq, write K suppphq and fix some compact unit neighborhood U in H. 25 / 30

34 density formula For any g P KLpHq such that supppgq Ď KU we then have for n sufficiently large the estimate ˇ hpxqdθ H pxq ωhpa n q H θ G pa n qˇ ď ˇ hpxqdθ H pxq gpxqdθ H pxq ˇ ` H H ` ˇ gpxqdθ H pxq ωg pa n q H θ G pa n qˇ ` ω g pa n q ˇθ G pa n q ωhpa n q θ G pa n qˇ ď h g 8 `θh pkuq ` 2 denspλq ` ˇ gpxqdθ H pxq ωg pa n q θ G pa n qˇ. Since KLpHq is dense in C c phq we find g P KLpHq (of support contained in KU) such that the first term in the above estimate does not exceed ε{2. by the density formula for KLpHq the second term is also smaller than ε{2 if n is sufficiently large. H 26 / 30

35 diffraction regular model sets are pure point diffractive Theorem Let pg, H, Lq be a cut-and-project scheme, and let h 1 W for W Ă H relatively cpct measurable and θ H pbw q 0. Then the weighted model set ω h has autocorrelation γ and diffraction pγ given by γ ω h r h, pγ ω q h 2 history Hof 95 via harmonic analysis (euclidean G, H) Schlottmann 00 via dynamical systems Baake Moody 04 via almost periodic measures R Strungaru via PSF 27 / 30

36 diffraction proof autocorrelation γ of ω h vague limit of finite ac measures where An η n pzq γ n 1 θpa n q ω h Bn Čω h Bn ÿ η n pzqδ z, zpl denotes restriction w.r.t. any van Hove pa n q npn and 1 θ G pa n q ÿ xpnpw XpW `z qqxa n hpx qhpx z q lim nñ8 η n pzq h rhpz q for all z P L by density formula as supppγq uniformly discrete, γ n converges to ω h r h since h rh P KLpHq, transform follows from PSF 28 / 30

37 diffraction diffraction of the Fibonacci chain pγ ÿ kpzrτs{? 5 ˆ τ? 5 2 ˆsinpπτk q πτk 2 δ k Zrτs tm ` nτ m, n P Zu L peaks dense in p G R! star map: pm ` nτq m n{τ note that pω does not exist as a measure since sinpxq{x is not an L 1 function. 29 / 30

38 diffraction references L.N. Argabright and J. Gil de Lamadrid, Fourier analysis of unbounded measures on locally compact abelian groups, Memoirs of the Amer. Math. Soc. 145 (1974). C. Berg and G. Forst, Potential theory on locally compact abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 87, Springer (1975). A. Deitmar and S. Echterhoff, Principles of harmonic analysis, Springer (2009). R.V. Moody, Meyer sets and their duals, in: The mathematics of long-range aperiodic order (ed R.V. Moody), NATO ASI Series C489, Kluwer (1997), A. Hof, On diffraction by aperiodic structures, Comm. Math. Phys. 169 (1995), M. Baake and R.V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. 573 (2004), C. Richard and N. Strungaru, Poisson summation and pure point diffraction (2015), in preparation. 30 / 30