Quantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions

p.1 Quantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions ICIAM07, July 18, 2007 Alex Barnett Mathematics Department, Dartmouth College

p.2 PlanarDirichleteigenproblem Normalmodesofelasticmembraneor drum (Helmholtz,Germain,19 th C) Eigenfunctions φ j oflaplacian := x 2 1 + x 2 2 inboundedcavity Ω R 2 φ j = E j φ j φ j Ω = 0 DirichletBC φ Ω iφ j dx = δ ij

PlanarDirichleteigenproblem Normalmodesofelasticmembraneor drum (Helmholtz,Germain,19 th C) Eigenfunctions φ j oflaplacian := x 2 1 + x 2 2 inboundedcavity Ω R 2 φ j = E j φ j φ j Ω = 0 DirichletBC φ Ω iφ j dx = δ ij mode j = 1 discrete eigenvalues E 1 <E 2 E 3 wavenumber k j =E 1/2 j wavelength λ j := 2π k j Time-harmonic solns of wave eqn(acoustics, optics, quantum, etc) Asymptoticsof φ j aseigenvalue E j?dependsonshape... p.2

p.3 (Somefavorite)eigenmodetopics I. Background, chaotic billiards II. Quantum ergodicity: how uniform are eigenmodes? III. Scarring and the mushroom: how do periodic ray orbits affect mode statistics?

p.4 Somehistoryof(related)eigenmodes Ernst Chladni(1756 1827) sprinkles sand on metal plates Plays themwithviolinbow:visualizesnodallines(φ j = 0)

p.4 Somehistoryof(related)eigenmodes Ernst Chladni(1756 1827) sprinkles sand on metal plates Plays themwithviolinbow:visualizesnodallines(φ j = 0) popular lectures all over Europe Napoleon impressed: offers 1 kg gold to explain patterns Napoleon realised: irregularly shaped plate harder to understand (first funding for quantum chaos!) Sophie Germain got prize in 1816 Note: rigid plate membrane (biharmonic 2 ratherthanlaplacian )

p.5 Einstein-Brillouin-Keller(EBK) (Keller 60) Recallinseparabledomains(ellipse,rect.): φ j productsof1dmodes

p.5 Einstein-Brillouin-Keller(EBK) (Keller 60) Recallinseparabledomains(ellipse,rect.): φ j productsof1dmodes EBK: semiclassical approx. for certain non-separable modes φ(x) M m=1 A m(x)e iks m(x) mode sum of traveling waves, rays S m =phasefunction A m =amplitudefunction Insertinto φ = Eφ gives S m = 1 phasegrowsalongstraightrays For single-valued φ to exist and satisfy BCs: i) rays reflect off boundary, giving ray families which must close ii)quantization:round-tripphase = 2πn + π (#focalpoints) + π(#reflections) 2

p.6 Bouncingrays:thegameofbilliards x Ω ξ θ i θ θ r fullphasespace (x 1,x 2,ξ 1,ξ 2 ) =: (x,ξ) free motion: Hamiltonian dynamical system energy H(x,ξ) = ξ 2 conserved trajectory x(t)launchedat x(0) = x 0 TWO BROAD CLASSES OF MOTION 1 0.9 integrable: ergodic: 0.8 0.7 0.6 0.5 0.4 0.3 0.2 d conserved quantities(d=2) 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 only H conserved: chaos!

p.7 Propertiesofchaoticrays Defn.ofergodic:Given A(x)testfunc,fora.e.trajectory x 0, lim T 1 T T 0 time average = spatial average A(x(t))dt = 1 vol(ω) Ω A(x)dx =: A

p.7 Propertiesofchaoticrays Defn.ofergodic:Given A(x)testfunc,fora.e.trajectory x 0, lim T 1 T T 0 time average = spatial average A(x(t))dt = 1 vol(ω) Ω A(x)dx =: A hyperbolic : x 1 (t) x 2 (t) ce Λt 0 < Λ =Lyapunovexponent Anosov : uniformly hyperbolic (all periodic orbits unstable, isolated) e.g. Bunimovich stadium ergodic but not Anosov... QUANTUM CHAOS : study of eigenmodes when Ω ergodic(einstein 1917) Apps: quantum dots, nano-scale devices, molecular phys/chem Why chaos important? Generic shapes have some chaotic phase space Modes φ j irregular: VIEW j 3000:45wavelengthsacross...saymore?

p.8 II.QuantumErgodicityTheorem(QET) studymodematrixelements φ j, Aφ j := Ω A(x) φ j(x) 2 dx

p.8 II.QuantumErgodicityTheorem(QET) studymodematrixelements φ j, Aφ j := Ω A(x) φ j(x) 2 dx Thm:if Ωergodic,a.e.matrixelementtendstothemean: lim φ j,aφ j A = 0 E j jexceptsubseq.ofvanishingdensity (Schnirelman 74, Colin de Verdière 85, Zelditch 87, Z-Zworski 96) measure φ j 2 dx weaklyin L1 dx/vol(ω) j = 1 j = 10 j = 10 2 j = 10 3 j 5 10 4

p.9 QuantumUniqueErgodicity(QUE) Conj.(Rudnick-Sarnak 94): There is no excluded subsequence in QET

p.9 QuantumUniqueErgodicity(QUE) Conj.(Rudnick-Sarnak 94): There is no excluded subsequence in QET context: negatively curved manifolds(anosov) Recent analytic results: Proventoholdforarithmeticmanifold SL 2 (Z) \ H(Lindenstrauss 03)...veryspecialsystem,symmetries,all Λ = 1 ProvennottoholdforsomequantumArnoldcatmaps(Faureetal 03)

p.10 Numericalexperiments (B, Comm. Pure Appl. Math. 06) dispersing cavity, proven Anosov(Sinai 70) desymmetrized, generic Λ exponents { 1 in Ω A, testfunction A = 0 otherwise

p.10 Numericalexperiments (B, Comm. Pure Appl. Math. 06) dispersing cavity, proven Anosov(Sinai 70) desymmetrized, generic Λ exponents { 1 in Ω A, testfunction A = 0 otherwise Large-scalestudy,30,000modesinrange j 10 4 to 10 6,enabledby: 1. Efficientboundary-basednumericsfor φ j ( scalingmethod ) 2. Matrixelements Ω A φ 2 jdxviaboundaryintegralson Ω A 100timeshigherin jthananypreviousstudies(e.g.bäcker 98) only a few CPU-days total

p.11 Typicalhigh-frequencyergodicmode 225 wavelengths across system level number j 5 10 4 E j 10 6 Stringiness unexplained...

(compare: random sum of plane waves) Re ikm x a e m m P all wavenumbers km = E = const. similarly stringy... interesting to the eye only? p. 12

Raw matrix element data To reach high E, only use modes in intervals Ej [E, E + L(E)] diagonal elements classical mean 0.65 <φn, A φn> 0.6 0.55 0.5 0.45 0.6 0.4 0.5 0.4 0.35 4 5 10 0 2 4 6 10 E 6 n 7 10 8 10 10 5 x 10 No outliers strong evidence for QUE (exceptional density < 3 10 5 ) p. 13

p.14 Atwhatratecondensetothemean? interval I E = [E,E + L(E)] choosewidth L(E) = O(E 1/2 ) eigenvaluecountininterval N(I E ) := #{j : E j I E } quantumvariance V A (E) := 1 φj,aφ j A 2 N(I E ) E j I E Random-wavestatisticalmodelfor φ j gives: V A (E) a RW E 1/2 Conj.(Feingold-Peres 86): V A (E) g C A (0) vol(ω) E 1/2 where g = 2fromstatisticalindependenceofnearby φ j (heuristic) C A (ω) :=FTofautocorrelation 1 vol(ω) Ω A(x 0)A(x(τ))dx A 2

p.15 Resultsonquantumergodicityrate measured V A (E) 10 3 Conj. 1.2 (best fit) random wave model Conj. 1.3 (FP) consistent with power law model V A (E) = ae γ V A (E) 1.2 1.1 Ratio V A (E) / FP fit γ = 0.48 ± 0.01 10 4 1 0.9 0.8 10 5 10 6 10 7 10 4 10 5 10 6 10 7 E verycloseto γ = 1/2 prefactor: RW model fails FP Conj. succeeds large numbers of modes unprecedented accuracy(< 1%) asymptotic regime seen for first time(but more data needed!) consistentwithfpconj.,convergenceveryslow:7%offat j = 10 5

Note also exceptional bouncing ball BB modes(since not Anosov) Apps of scars: dielectric micro-lasers(tureci et al), tunnel diodes... p.16 III.Scars:shadowsofperiodicrayorbits Somehigh-jmodes φ j 2 localizeonunstableperiodicorbits(upo) discovered by numerical study of quarter-stadium modes (Heller 84)

p.17 Whydomodesshowscars? Statistics of modes Greens function wave propagation

p.17 Whydomodesshowscars? Statistics of modes Greens function wave propagation Indistributionalsensein k(wavenumber),for x Ω, j=1 φ j (x) 2 δ(k k j ) = 2k π lim ε 0 Im G(x,x;k2 + iε) Local Density Of States Greens func for Helmholtz eqn in Ω Then use semiclassical ray theory(heuristic): G(x,x;k 2 ) (freespace) + returning orbits x x (geometricfactor) e ik(orbitlength)

Whydomodesshowscars? Statistics of modes Greens function wave propagation Indistributionalsensein k(wavenumber),for x Ω, j=1 φ j (x) 2 δ(k k j ) = 2k π lim ε 0 Im G(x,x;k2 + iε) Local Density Of States Greens func for Helmholtz eqn in Ω Then use semiclassical ray theory(heuristic): G(x,x;k 2 ) (freespace) + returning orbits x x (geometricfactor) e ik(orbitlength) Outcome: k-periodic LDOS enhancement along each periodic orbit (Gutzwiller, Heller, Bogomolny, Berry, Kaplan, 80-90s) p.17

Mushroomcavitymodes (B-Betcke, nlin/0611059) Unusually simple divided phase space (Bunimovich 01) ergodic rays regular rays p.18

Mushroomcavitymodes (B-Betcke, nlin/0611059) Unusually simple divided phase space (Bunimovich 01) First calculation of high-freq modes: j 2000 ergodic rays regular rays Percival 73 conjecture verified(percival 73): modes localize to either regular or chaotic region p.18

Veryhighfreqmushroommodes ergodic, equidistributed ergodic, strongly scarred p.19

Newphenomenon:a movingscar n φ j (q) 2,boundarylocation q: 600 500 400 j 300 200 100 0 1 2 3 4 5 q sloping streak effect MODES SMOOTHED p.20

Newphenomenon:a movingscar n φ j (q) 2,boundarylocation q: 600 takeftalong kaxis,gives: wave autocorrelation in time 500 18 16 14 400 12 l 10 j 300 8 6 4 200 100 2 0 0 1 2 3 4 5 q 0 1 2 3 4 5 q sloping streak effect MODES SMOOTHED refocused returning orbits return length varies with q EVOLVE p.20

p.21 Conclusion Highfrequencyasymptoticpropertiesofchaoticmodes φ j : φ j (x) 2 tendstospatiallyuniformatconjecturedrate Scarring on unstable periodic orbits, enhanced by refocusing Topics not covered: Bouncing ball mode leakage Quasi-orthogonalityofboundaryfunctions n φ j (s) (ongoing w/ A. Hassell) Findouthowthenumericalmethodworks:Friam:KOLF121 Thanks: P. Deift(NYU) A. Hassell(ANU) P. Sarnak(Princeton) T. Betcke(Manchester) S. Zelditch(JHU) Funding: NSF(DMS-0507614) Preprints, talks, movies: http://math.dartmouth.edu/ ahb madewith:linux,l A TEX,Prosper

p.22 High-eigenvaluequarter-stadiumscar Our numerical evidence for QUE scars die, measure o(1)

Husimidistributionsonmushroomboundary 1 0.5 ergodic integrable classical boundary PSOS: p 0 0.5 ergodic 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 q c q L 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 p 0 p 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1 2 3 4 5 q 1 2 3 4 5 q regular ergodic, strongly scarred p.23

IV.BouncingBallmodes (ongoing work w/ A. Hassell) p.24