Wavefunctions in chaotic quantum systems Arnd B acker Institut f - PowerPoint PPT Presentation

IV SoE – Random wave model – consequences Amplitude distribution is Gaussian For the amplitude distribution P n ( ψ ) of an eigenfunction ψ n ( q ) � b vol( { q ∈ Ω | ψ n ( q ) ∈ [ a, b ] ⊂ R } ) =: P n ( ψ ) d ψ . (24) vol(Ω) a the random wave model implies � � − ψ 2 1 √ P ( ψ ) = 2 πσ exp . (25) 2 σ 2 with variance σ 2 = 1 / vol(Ω) . Arnd B¨ acker 26 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Random wave model – consequences Amplitude distribution – example 0.5 P( ψ ) 0.4 0.3 0.2 0.1 0.0 ψ -4 -3 -2 -1 0 1 2 3 4 Arnd B¨ acker 27 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Random wave model – consequences 2 Bound on the growth of eigenfunctions For random waves one has with probability one (see ([R. Aurich, AB, R. Schubert, and M. Taglieber ’99]) ) √ max x ∈ Ω | f ( x ) | √ lim sup ≤ 3 2 . (26) ln E E →∞ In contrast to the general result ([Seeger, Sogge ’89; Grieser ’97]) || ψ n || ∞ < c E 1 / 4 , (27) n which is sharp (e.g. sphere S 2 , circle billiard). Arnd B¨ acker 28 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Maximum norms – (some) known results • Conjecture [Sarnak 95, Iwaniec and Sarnak 95] : for surfaces of constant negative curvature: || ψ n || ∞ < c ε E ε ∀ ε > 0 (28) , . n (related to Lindel¨ of hypothesis) • Arithmetic surfaces [Iwaniec and Sarnak 95] : for a Hecke basis 5 24 + ε || ψ n || ∞ < c ε E ∀ ε > 0 , . n � || ψ n j || ∞ ≥ c ln ln E n j , for a subsequence . • Arithmetic three manifolds [Rudnick and Sarnak 94, Koyama 95] : there exist || ψ n || ∞ < c ε E 37 / 76+ ε systems with , n || ψ n j || ∞ > cE 1 / 4 and a system with , n j for Hecke eigenfunctions. Arnd B¨ acker 29 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Maximum norms – Euclidean billiards Stadium billiard 2000 odd-odd eigenfunctions 2000 even-even eigenfunctions 8 8 e) f) L ∞ L ∞ 6 6 4 4 2 2 0 0 0 10000 20000 E 0 10000 20000 E Cardioid: 6000 odd eigenfunctions Circle billiard: 1244 eigenfunctions 8 8 L ∞ d) L ∞ 6 6 4 4 2 2 0 0 0 20000 40000 60000 E 0 5000 10000 E 15000 Arnd B¨ acker 30 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Maximum norms – constant negative curvature Arithmetic triangle: 2099 functions Non-arithmetic triangle: 2092 functions 8 8 a) b) L ∞ L ∞ 6 6 4 4 2 2 0 0 0 10000 20000 E 0 10000 20000 E 3139 eigenfunctions Octagon 500 eigenfunctions 8 8 c) L ∞ L ∞ 6 6 4 4 2 2 0 0 0 10000 20000 30000 E 40000 315000 317500 E 320000 blue – maxima of eigenfunctions red – mean of maxima of 200 random waves Arnd B¨ acker 31 � ⇓ ⇐ ⇒ Σ ⊕

IV SoE – Maximum norms – eigenstates with large norm Arnd B¨ acker 32 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity Semiclassical eigenfunction hypothesis [Berry ’77, ’83, Voros ’79] The Wigner function � 1 n ( q − q ′ / 2) ψ n ( q + q ′ / 2) d q ′ , e i pq ′ ψ ∗ W n ( p , q ) := (2 π ) 2 semiclassically concentrates on those regions in phase space, which a typical orbit explores in the long time limit t → ∞ . Implications for integrable systems chaotic systems Arnd B¨ acker 33 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity Consequences of the semiclassical eigenfunction hypothesis: For integrable systems : Localization on invariant tori W ( p , q ) ∼ δ ( I ( p , q ) − I ) (29) (2 π ) 2 (here: I ( p , q ) : action variable) For chaotic systems 1 W n ( p , q ) → vol(Σ E ) δ ( H ( p , q ) − E ) , (30) i.e. semiclassical condensation on the energy surface Σ E . Remark: for ergodic systems one can show ([AB, RS, PS ’98]) QET implies the semiclassical eigenfunction hypothesis (when restricted to a subsequence of density one). Arnd B¨ acker 34 � ⇓ ⇐ ⇒ Σ ⊕

V QE — observables and operators [AB, R. Schubert, P. Stifter ’98] Classical observables are functions on phase space R 2 × Ω , The mean value of an observable a ( p , q ) at energy E is given by � 1 a E = a ( p , q ) d ν . (31) vol(Σ E ) Σ E Weyl symbol W[ A ] : To an operator A associate � � � q − q ′ 2 , q + q ′ e i q ′ p K A d 2 q ′ , W[ A ]( p , q ) := (32) 2 R 2 where K A is the Schwarz kernel, Aψ ( q ) = K A ( q , q ′ ) ψ ( q ′ ) d 2 q ′ . � Ω A is called a pseudodifferential operator , A ∈ S m (Ω) , if its Weyl symbol belongs to a certain class of functions S m ( R 2 × Ω) ⊂ C ∞ ( R 2 × Ω) . Arnd B¨ acker 35 � ⇓ ⇐ ⇒ Σ ⊕

V QE — observables and operators Weyl quantization: a �→ A To any function a ∈ S m ( R 2 × Ω) one can associate an operator Op [ a ] ∈ S m (Ω) , � � �� p , q + q ′ 1 e i ( q − q ′ ) p a f ( q ′ ) d 2 q ′ d 2 p Op [ a ] f ( q ) := (2 π ) 2 2 Ω × R 2 such that its Weyl symbol is a , i.e. W[ Op [ a ]] = a . cl ( R 2 × Ω) ⊂ C ∞ ( R 2 × Ω) : Classical symbols S m have an asymptotic expansion in homogeneous functions in p , ∞ � a m − k ( p , q ) , with a m − k ( λ p , q ) = λ m − k a m − k ( p , q ) a ( p , q ) ∼ k =0 Arnd B¨ acker 36 � ⇓ ⇐ ⇒ Σ ⊕

V QE — observables and operators Classical pseudodifferential operators : S m cl : corresponding class of pseudodifferential operators m ∈ R : order of the pseudodifferential operator. cl (Ω) and W[ A ] ∼ � ∞ Principal symbol: For A ∈ S m k =0 a m − k the leading term a m ( p , q ) is called the principal symbol of A The principal symbol denoted by σ ( A )( p , q ) := a m ( p , q ) . For details see e.g.: • [AB, R. Schubert, P.Stifter ’98] • [R. Schubert 2001] Arnd B¨ acker 37 � ⇓ ⇐ ⇒ Σ ⊕

V QE — observables and operators The Wigner function of a state | ψ � is given as the Weyl symbol of the corresponding projection operator | ψ �� ψ | � � � � � q − q ′ q + q ′ � � e i q ′ p ψ ⋆ d 2 q ′ . | ψ �� ψ | ( p , q ) = ψ W 2 2 R 2 (33) From the Wigner function one can recover | ψ ( q ) | 2 by � � � 1 | ψ ( q ) | 2 = ( p , q ) d 2 p . W | ψ �� ψ | (34) (2 π ) 2 R 2 For the expectation value � ψ, Aψ � we have �� � � 1 ( p , q ) d 2 p d 2 q . � ψ, Aψ � = | ψ �� ψ | W[ A ]( p , q ) W (2 π ) 2 Ω × R 2 Arnd B¨ acker 38 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity theorem QET [Shnirelman ’74, Colin de Verdi` ere ’85, Zelditch ’87, Zelditch/Zworski ’96, ....] For ergodic systems there exists a subsequence { n j } of density one such that j →∞ � ψ n j , Aψ n j � = σ ( A ) , lim (35) for every classical pseudodifferential operator A of order zero. Here σ ( A ) is the principal symbol of A . And σ ( A ) is its classical expectation value, �� 1 a ( p , q ) δ ( p 2 − 1) d p d q . a = (36) vol(Σ 1 ) R 2 × Ω # { n j | E n j < E } A subsequence { n j } ⊂ N has density one if lim = 1 , N ( E ) E →∞ where N ( E ) := # { n | E n < E } is the spectral staircase function. Arnd B¨ acker 39 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity theorem – Special Case Classical ergodicity of a flow { φ t } � T 1 χ D ( φ t ( p , q ) d t = vol( D ) lim 2 T vol(Ω) T →∞ − T for almost all initial conditions in phase space, ( p , q ) ∈ T ∗ Ω . Quantum ergodicity in position space � χ D ( q ) | ψ n j ( q ) | 2 d 2 q = vol( D ) lim vol(Ω) j →∞ Ω for a subsequence of density one. Quantum ergodicity theorem makes statement about sequences of eigenfunctions (weak limit!). Arnd B¨ acker 40 � ⇓ ⇐ ⇒ Σ ⊕

V QET – Example – observable in position space Consider as observable A = χ D ( q ) and plot � χ D ( q ) | ψ n j ( q ) | 2 d 2 q − vol( D ) (37) vol(Ω) Ω 0.03 d(n) 0.02 0.01 0.00 -0.01 -0.02 -0.03 0 1000 2000 3000 4000 5000 n 6000 Arnd B¨ acker 41 � ⇓ ⇐ ⇒ Σ ⊕ Quite strong fluctuations around 0.

✩ ✫ � ✕ ✂ ✝ ✩ ☞ ✕ ✌ ✍ ☞ � ✌ ☞ ✟ ✔ ✕ ✖ ✙ ✕ ✕ � ✁ V QET – Example – observable in position space Thus consider cumulative differences � � � � � � � � 1 χ D ( q ) | ψ n j ( q ) | 2 d 2 q − vol( D ) � � S 1 ( E, A ) := . � � N ( E ) � � vol(Ω) � � n : E n ≤ E Ω ✩✮✭✯✩✰✔✘✩ ✟✏✎✑✟ ☞✒✌ ✄✆☎ ☞✘✗✚✙✜✛✠✢✤✣✤✥✧✦ ✟✏✎✑✟ ✩✮✭✯✩ ✟✏✎✑✟☛✟ ✟✏✎✑✟☛✟✓✞ ✩✮✭✯✩ ✟☛✟✡✟ ✟☛✟☛✟ ✟☛✟☛✟ ✟☛✟☛✟☛✟ ✞✠✟✡✟☛✟☛✟ ✩✮✭✯✩✪✩✰✫ ✩✮✭✯✩✪✩✪✩ ✫✆✩✪✩✪✩ ✩✪✩✪✩✬✩ ✫✘✩✬✩✪✩ ✔✆✩✪✩✪✩✪✩ ✔✪✫✆✩✪✩✪✩ ★✪✩✪✩✪✩✪✩ Remark: QET is equivalent to S 1 ( E, A ) → 0 as E → ∞ . Arnd B¨ acker 42 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity theorem — “example” 1 Example: Square billiard (to confuse you ... ;-): ψ kl ( x, y ) = 1 π sin( kx ) sin( ly ) k, l ∈ N (38) Then one gets �� | ψ kl ( x, y ) | 2 d x d y D 2 π D Ω y 1 � x 1 � 2 π = 1 d y sin 2 ( kx ) sin 2 ( ly ) d x (39) π 2 x 0 y 0 → ( x 1 − x 0 )( y 1 − y 0 ) ≡ vol( D ) (40) π 2 vol(Ω) for a subsequence of density one. Arnd B¨ acker 43 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity theorem — “example” 2 Consider the observable a ( p , q ) = a ( p ) . Then � | � ψ n ( p ) | 2 a ( p ) d 2 p , � ψ n , Aψ n � = (41) R 2 with � 1 � e i pq ψ ( q ) d 2 q ψ n ( p ) = (42) (2 π ) 2 R 2 Characteristic function in momentum space a ( p ) = χ C ( θ,δθ ) ( p ) where � � ( r, ϕ ) | r ∈ R + , ϕ ∈ [ θ − δθ/ 2 , θ + δθ/ 2] C ( θ, δθ ) := (43) Arnd B¨ acker 44 � ⇓ ⇐ ⇒ Σ ⊕

V Quantum ergodicity theorem — “example” 2 QET implies for a subsequence of density one: �� ψ n j ( p ) | 2 d 2 p = δθ | � lim (44) 2 π n j →∞ C ( θ,δθ ) Example: Circle billiard (to confuse you even more ... ;-): ψ kl ( r, φ ) = J k ( j kl r ) cos( kφ ) (45) One can show that a subsequence of density one of eigenfunctions is quantum ergodic in momentum space. Arnd B¨ acker 45 � ⇓ ⇐ ⇒ Σ ⊕

V QET – questions Several interesting questions Do exceptional eigenfunctions exist ? E.g.: scars, bouncing ball modes, . . . (quantum limit has to be invariant under the flow!) If yes, how many are there ? The quantum ergodicity theorem implies N exceptional ( E ) = 0 . lim N ( E ) E →∞ Can one say more about N exceptional ( E ) ? How fast do quantum expectation values tend to the corresponding classical limit ? I.e., what is the rate of quantum ergodicity ? Arnd B¨ acker 46 � ⇓ ⇐ ⇒ Σ ⊕

V QET — quantum limits Quantum limits (in position space) Consider the sequence of probability measures on Ω d µ n := | ψ n ( q ) | 2 d 2 q (46) Definition A measure µ ql is called quantum limit if a subse- quence of the µ n converges to µ ql . QET: for a subsequence of density one the quantum limit is d µ = d 2 q . (47) What quantum limits can occur? They have to be invariant under the flow! Arnd B¨ acker 47 � ⇓ ⇐ ⇒ Σ ⊕

V QET – quantum limits (sketch of invariance) For a quantum limit µ ql consider � � a ( q ) | ψ n j | 2 d 2 q → a ( q ) d µ ql � ψ n j , Aψ n j � = (48) Ω Ω We have (where U t is the time evolution operator) � ψ n j , Aψ n j � = � ψ n j , U − t AU t ψ n j � (49) Next we use Theorem (Egorov, special case) Under certain assumptions � � U ∗ = σ ( A ) ◦ φ t σ t AU t (50) I.e.: time evolution for finite times and quantization commute in the semiclassical limit. Arnd B¨ acker 48 � ⇓ ⇐ ⇒ Σ ⊕

V QET – quantum limits (sketch of invariance) For � ψ n j , Aψ n j � = � ψ n j , U − t AU t ψ n j � (51) the Egorov theorem gives � ψ n j , U − t AU t ψ n j � = � ψ n j , Op ( a ◦ φ t ) , ψ n j � + lower order terms From this: quantum limits are invariant under the flow. Possible examples: Liouville measure unstable periodic orbits collection of finitely/countably many unstable periodic orbits marginally stable orbits (eg stadium billiard) and: combinations of these Arnd B¨ acker 49 � ⇓ ⇐ ⇒ Σ ⊕

V QET – eigenfunctions cardioid Arnd B¨ acker 50 � ⇓ ⇐ ⇒ Σ ⊕

V QET – eigenfunctions stadium Arnd B¨ acker 51 � ⇓ ⇐ ⇒ Σ ⊕

V QET – exceptional eigenfunctions Look at sequence of eigenfunctions in the cardioid billiard . . . . . . there are states, localizing around unstable periodic orbits ( “scars” ) And for the stadium billiard . . . . . .“Bouncing–Ball–Modes” : Arnd B¨ acker 52 � ⇓ ⇐ ⇒ Σ ⊕

V QET – BBMs – Counting function Quantum limit for bouncing ball modes: In position space n j →∞ supp ( ψ n j ) ⊂ Ω B lim (52) and in momentum space ψ n j | 2 = δ ( p x ) δ ( p y − 1) + δ ( p y + 1) n j →∞ | � lim (53) 2 Consider counting function � � N bb ( E ) := n | ψ n is a bouncing ball mode (54) The QET implies for E → ∞ N bb ( E ) N ( E ) → 0 (55) Arnd B¨ acker 53 � ⇓ ⇐ ⇒ Σ ⊕

V QET – BBMs – Counting function One can show ( [G. Tanner ’97] , [AB, R. Schubert, P. Stifter ’97] ) Stadium billiard y N bb ( E ) ∼ cE 3 / 4 L 0 B(y) Cosine billiard L(x) N bb ( E ) ∼ cE 9 / 10 x B 0 B 1 L ( x ) ∼ L 0 − C ( B 0 + x ) γ δ = 1 1 Remark: 2 + 2 + γ . For every 1 2 < δ < 1 one can find an ergodic Sinai billiard, s.t. N bb ( E ) ∼ cE δ . This suggests: the QET is sharp Recent results: [Burq,Zworski 2003], [Zelditch 2003] Arnd B¨ acker 54 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function Counting function for bouncing ball modes, stadium billiard 250 N bb (E) 200 150 100 Fit to αE δ α = 0 . 20 50 δ = 0 . 76 0 E 0 2000 4000 6000 8000 10000 Arnd B¨ acker 55 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function Counting function for bouncing ball modes, cosine billiard 150 N bb (E) 120 90 60 Fit to αE δ + β α = 0 . 04 30 δ = 0 . 87 β = 12 . 4 0 8000 E 0 2000 4000 6000 Arnd B¨ acker 56 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function A second look at a sequence of bbm’s: Arnd B¨ acker 57 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function Linear superposition ψ 321 ψ 322 � �� � “+” “ − ” sin(0 . 2 π ) ψ 321 − cos(0 . 2 π ) ψ 322 cos(0 . 2 π ) ψ 321 + sin(0 . 2 π ) ψ 322 Arnd B¨ acker 58 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function Parameter variation 1700 E 1690 A’ 1680 B’ E’ F’ A B E F 1670 1.78 1.79 1.80 1.81 a 1.82 Arnd B¨ acker 59 � ⇓ ⇐ ⇒ Σ ⊕

V BBMs – Counting function A A’ B B’ C C’ D’ D E E’ F F’ Arnd B¨ acker 60 � ⇓ ⇐ ⇒ Σ ⊕

V QET – quasimodes (Arnold ’72) Definition A pair ( ˜ ψ, ˜ E ) , where ˜ ψ : Ω → R and ˜ E ∈ R , is called quasimode with discrepancy ǫ if || ∆ ˜ ψ + ˜ E ˜ ψ || < ǫ , where � || · || := �· , ·� Proposition (Lazutkin ’93) • The interval [ ˜ E − ǫ, ˜ E + ǫ ] contains at least one eigenvalue of − ∆ . • If there is only one eigenvalue E n with eigenfunction ψ n in this interval, then || ˜ ψ − ψ n || < Cǫ . I.e. the quasimode is an approximate eigenfunction. • If there is more than one eigenvalue in this interval � ˜ ψ ( q ) ≈ a n ψ n ( q ) . (56) E n ∈ [ ˜ E − ǫ, ˜ E + ǫ ] Arnd B¨ acker 61 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Basic idea: Scars are eigenfunctions showing an enhanced density around an unstable periodic orbit Theoretical studies: • Heller ’84, Kaplan/Heller ’98 • Bogomolny ’88, Berry ’89 • Ozorio de Almeida ’98 • ... many others ... Problems • not really a definition • not constant in time ;-) Arnd B¨ acker 62 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Some more details Expectation: scars should occur at around energies ) 2 where E scar = ( k scar n n � � = 2 π n + ν γ k scar (57) n l γ 4 Plot of a scar measure: (via Poincar´ e Husimi function) 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 2000 2200 2400 2600 2800 3000 3860 3880 3900 3920 3940 3960 Arnd B¨ acker 63 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Eigenfunctions in the cluster: Arnd B¨ acker 64 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars difference to k scar Energies of scars: n 250 30 k n E n -E scar,n 20 200 10 0 150 -10 100 -20 -30 50 -40 0 -50 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 n n Quite strong fluctuations !! (mean spacing: 4 π 3 π/ 4 = 16 4 π A = 3 = 5 . 333 . . . ) Arnd B¨ acker 65 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Remark: For surfaces of constant negative curvature: no scars observed, see Aurich, Steiner ’95 Auslaender, Fishman ’98 Possible types of scars: (simplified) “super-strong scarring” : quantum limit is a δ function on a periodic orbit strong scarring: quantum limit is a δ function on a periodic orbit + Liouville measure “soft scarring” : quantum limit is the Liouville measure Arnd B¨ acker 66 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Some results on this: • For any Anosov map on the torus: weight for scars (on a √ finite union of periodic orbit): < ( 5 − 1) / 2 ([F. Bonechi, S. De Bi´ evre 2003]) • Explicit construction of a sequence of states for the cat map for which the quantum limit is the sum of 1 / 2 Lebesgue + 1 / 2 δ on any periodic orbit. ([F. Faure, S. Nonnenmacher, S. De Bi` evre 2003]) • weight for scars (on a finite or countable union of p.o.): < 1 / 2 , ([F. Faure, S. Nonnenmacher 2003]) Arnd B¨ acker 67 � ⇓ ⇐ ⇒ Σ ⊕

V QET — Scars Quantum unique ergodicity: • proven for ergodic linear parabolic maps on T 2 ([Marklof, Rudnick 2000]) • for certain cat maps: QUE for joint eigenstates with Hecke operators, ([Rudnick, Kurlberg 2000]) (not all eigenstates are of this type) • for sequences of joint eigenstates of the Laplacian and Hecke operators on arithmetic surfaces ([Lindenstrauss 2003]) (all eigenstates are conjectured to be of this type) Other extreme: • class of ergodic piecewise affine transformation on T 2 : all classical invariant measures appear as quantum limits. ([C. Chang, T. Kr¨ uger, R. Schubert, S. Troubetzkoy]) Arnd B¨ acker 68 � ⇓ ⇐ ⇒ Σ ⊕

V QET and random wave model Amplitude distribution revisited 0.5 P( ψ ) 0.4 0.3 0.2 0.1 0.0 -4 -3 -2 -1 0 1 2 3 ψ 4 2.0 P( ψ ) 1.5 1.0 0.5 0.0 Analytical expression available: ψ -4 -3 -2 -1 0 1 2 3 4 Arnd B¨ acker 69 � ⇓ ⇐ ⇒ Σ ⊕

V QET and random wave model Amplitude distribution for scars 0.5 P( ψ ) 0.4 0.3 0.2 0.1 0.0 ψ -4 -3 -2 -1 0 1 2 3 4 BBM and scars: Modification of the random wave conjecture Eigenfunctions of classically chaotic systems behave like random waves, but in general only for a subsequence of density one. Arnd B¨ acker 70 � ⇓ ⇐ ⇒ Σ ⊕

V Sketch: proof of the QET Theorem (Szeg¨ o limit theorem) For classical pseudodifferential operators one has � 1 lim � ψ n , Aψ n � = σ ( A ) (58) N ( E ) E →∞ E n ≤ E I.e. quantum mechanical mean values approach the classical mean. Theorem (Egorov, special case) Under certain assumptions � � U ∗ = σ ( A ) ◦ φ t σ t AU t (59) I.e.: time evolution for finite times and quantization commute in the semiclassical limit. Arnd B¨ acker 71 � ⇓ ⇐ ⇒ Σ ⊕

V Sketch: proof of the QET Consider now (following [R. Schubert, 2001] ) � � � 1 2 � � S 2 ( E, A ) := � � ψ n , Aψ n � − σ ( A ) . (60) � N ( E ) n : E n ≤ E Define � T A T := 1 U ∗ t [ A − σ ( A )] U t d t (61) T 0 for which (as U t | ψ n � = exp( − i tE n ) | ψ n � ) � ψ n , A T ψ n � = � ψ n , Aψ n � − σ ( A ) . (62) Thus � � � � 2 2 � � � � � � ψ n , Aψ n � − σ ( A ) = � � ψ n , A T ψ n � (63) � � ∗ A T ψ n � ≤ || A T ψ n || = � ψ n , A T (64) Arnd B¨ acker 72 � ⇓ ⇐ ⇒ Σ ⊕ Thus we get

V Sketch: proof of the QET With � 1 ∗ A T ψ n � S 2 ( E, A ) ≤ � ψ n , A T (66) N ( E ) n : E n ≤ E and the Szeg¨ o limit theorem we then obtain � 1 ∗ A T ) d µ E →∞ S 2 ( E, A ) ≤ lim σ ( A T (67) vol(Σ 1 ) Σ 1 � σ ( A T ) ∗ σ ( A T ) d µ = (68) Σ 1 Applying the Egorov theorem we have � T σ ( A T )( p, q ) = 1 σ ( A ) ◦ φ t ( p, q ) d t − σ ( A ) . (69) T 0 Arnd B¨ acker 73 As the flow φ t is ergodic we get � ⇓ ⇐ ⇒ Σ ⊕

V Sketch: proof of the QET Thus � � � 1 2 � � E →∞ S 2 ( E, A ) = lim lim � � ψ n , Aψ n � − σ ( A ) (71) � N ( E ) E →∞ n : E n ≤ E = 0 . (72) This is the mean value of a sequence of positive numbers. Thus there exists a subsequence { n j } of density one, such that n j →∞ � ψ n j , Aψ n j � = σ ( A ) lim (73) (see e.g. Walters) Moreover this holds for all A : diagonal argument, Zelditch ’87. Arnd B¨ acker 74 � ⇓ ⇐ ⇒ Σ ⊕

V QET – Summary Eigenfunctions in strongly chaotic systems: – Random wave model – Quantum ergodicity theorem: For ergodic systems: almost all eigenfunctions become equidistributed Possible exceptional eigenfunctions: bouncing ball modes, scars, . . . – QET = ⇒ semiclassical eigenfunction hypothesis (for ergodic systems, restricted to subsequence of density one) Not discussed – Rate of quantum ergodicity S 1 ( E, A ) = aE − 1 / 4 (?) – influence of non-quantum ergodic subsequences on the rate – Gaussian (?) flucutations of � ψ n , Aψ n � . Arnd B¨ acker 75 � ⇓ ⇐ ⇒ Σ ⊕

V Further topics Further topics Autocorrelation function and rate of quantum ergodicity Poincar´ e Husimi representation and quantum ergodicity Time evolution in chaotic systems Arnd B¨ acker 76 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg ([AB, R. Schubert 2002]) Local Autocorrelation function C loc ( q , δ x ) := ψ ∗ ( q − δ x / 2) ψ ( q + δ x / 2) . (74) In terms of the Wigner function � 1 n ( q − q ′ / 2) ψ n ( q + q ′ / 2) d q ′ , (75) e i pq ′ ψ ∗ W n ( p , q ) := (2 π ) 2 one has [Berry ’77] � W n ( p , q ) e i p δ x d p . C loc n ( q , δ x ) = (76) Arnd B¨ acker 77 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg For ergodic systems the quantum ergodicity theorem implies W n j ( p , q ) → δ ( H ( p , q ) − E n j ) , (77) vol(Σ E nj ) One gets for chaotic billiards in two dimensions [Berry ’77] √ 1 C loc ( q , δ x ) → vol(Ω) J 0 ( E | δ x | ) , (78) weakly as a function of q in the limit E → ∞ . Numerical tests (using a local average of C loc ( q , δ x ) ): Agreement is not too good — quite strong fluctuations Question: Can one understand/describe these results ? Arnd B¨ acker 78 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg P( Ψ ) 0.4 0.3 0.2 0.1 0 -4 -2 0 2 4 Ψ 1.0 θ =0 C(r, θ ) θ = π /4 θ = π /2 J 0 (r) 0.6 0.2 -0.2 -0.6 0 4 8 12 16 r 20 Arnd B¨ acker 79 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg Correlation length expansion �� √ E d p d q . ρ ( q − q ) W ( p , q ) e i p δ x / C ρ ( q , δ x ) = (79) For ρ = 1 one gets ∞ � ( − 1) l � J 2 l ( | δ x | ) + O ( E − 1 / 2 � C ( δ x ) = J 0 ( | δ x | ) + 2 π a 2 l cos(2 lθ ) + b 2 l sin(2 lθ ) l =1 where the coefficients a 2 l and b 2 l are the Fourier coefficients 2 π 2 π a 2 l = 1 b 2 l = 1 � � I ( ϕ ) cos(2 lϕ ) d ϕ I ( ϕ ) sin(2 lϕ ) d ϕ , (80) π π 0 0 of the radially integrated momentum density [ ˙ Zyczkowski ’92; AB, Schubert ’99] ∞ � | ˆ ψ ( r e ( ϕ )) | 2 r d r . I ( ϕ ) := (81) 0 Arnd B¨ acker 80 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg Relation to the rate of quantum ergodicity ∞ � ( − 1) l � J 2 l ( | δ x | ) + O ( E − 1 / 2 ) , � C ( δ x ) = J 0 ( | δ x | ) + 2 π a 2 l cos(2 lθ ) + b 2 l sin(2 lθ ) l =1 If the classical system is ergodic and ψ n j is a quantum ergodic sequence of eigenfunctions, then for j → ∞ � ψ n j , ˆ A 2 l ( q ) ψ n j � ∼ a 2 l = δ l 0 (82) � ψ n j , ˆ B 2 l ( q ) ψ n j � ∼ b 2 l = 0 . (83) Thus for E → ∞ we recover C ( r, θ ) = J 0 ( r ) . Deviations are determined by the rate of quantum ergodicity . Arnd B¨ acker 81 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg Removing the angular dependence As � 2 π 1 C ( r, θ ) d θ = J 0 ( r ) + O ( E − 1 / 2 ) , (84) 2 π 0 we consider the second moment, � 2 π 1 [ C ( r, θ ) − J 0 ( r )] 2 d θ . σ 2 ( r ) := (85) 2 π 0 Inserting the expansion of the autocorrelation function C ( δ x ) ∞ � 2 l )[ J 2 l ( r )] 2 (1 + O ( E − 1 / 2 )) . σ 2 ( r ) = 2 π 2 ( a 2 2 l + b 2 (86) l =1 Arnd B¨ acker 82 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg Second moment σ 2 ( r ) 2 π 1 Z [ C ( r, θ ) − J 0 ( r )] 2 d θ . σ 2 ( r ) := (87) 2 π 0 Expansion gives ∞ 2 l )[ J 2 l ( r )] 2 (1 + O ( E − 1 / 2 )) . X σ 2 ( r ) = 2 π 2 ( a 2 2 l + b 2 (88) l =1 0.04 0.0012 σ 2 (r) difference 0.0008 0.03 0.0004 0.0000 -0.0004 r 0 10 20 30 40 0.02 0.01 numeric expansion 0.00 0 20 40 60 80 r 100 Arnd B¨ acker 83 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg According to [Eckhardt et. al. ’95] we expect in the mean (under suitable conditions on the system) � � 2 ∼ 4 σ 2 � 1 cl ( A ) 1 � ψ n j , ˆ √ Aψ n j � − A (89) N ( E ) vol(Ω) E E n ≤ E for any pseudodifferential operator ˆ A of order zero with symbol A . √ Here A denotes the mean value of A , and σ cl ( A ) / T is the variance of the fluctuations of � 1 A ( p ( t ) , q ( t )) d t (90) T T around A . Arnd B¨ acker 84 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation function and rate of qerg Considering the mean of this function over all eigenfunctions up to energy E , and combining the previous � � � 2 ∼ 4 σ 2 1 cl ( A ) 1 � ψ n j , ˆ √ Aψ n j � − A (91) N ( E ) vol(Ω) E E n ≤ E and ∞ � 2 l )[ J 2 l ( r )] 2 (1 + O ( E − 1 / 2 )) . σ 2 ( r ) = 2 π 2 ( a 2 2 l + b 2 (92) l =1 we get � 1 σ 2 ( E, r ) := σ 2 n ( r ) (93) N ( E ) E n ≤ E ∞ � 8 π 2 � σ cl ( A 2 l ) 2 + σ cl ( B 2 l ) 2 � 1 [ J 2 l ( r )] 2 √ ∼ E . vol(Ω) l =1 Arnd B¨ acker 85 � ⇓ ⇐ ⇒ Σ ⊕

VI Autocorrelation fct and rate of qerg – Summary Origin of fluctuations around J 0 ( r ) : deviations from quantum ergodicity at finite energies Thus: Autocorrelation function allows to study the rate of quantum ergodicity! Remarks on σ 2 ( E, r ) : – Efficient quantity to measure the dependence of the rate of quantum ergodicity on different length scales. – For larger r ≡ | δ x | , one needs to incorporate higher terms in l which corresponds to expectation values of faster oscillating observables. Arnd B¨ acker 86 � ⇓ ⇐ ⇒ Σ ⊕

VII More recent results – Poincar´ e Husimi representation Question: Poincar´ e representation of eigenstates? AB, S. F¨ urstberger, R. Schubert: Poincar´ e Husimi representation of eigenstates in quantum billiards (2003) Natural starting point: normal derivative of the eigenfunction u n ( s ) := � ˆ n ( s ) , ∇ ψ n ( x ( s )) � , (94) Coherent states on the billiard boundary ∂ Ω � k � 1 / 4 � 2 σ ( s − q + mL ) 2 ] , e i k [ p ( s − q + mL )+ i c b ( q,p ) ,k ( s ) := (95) πσ m ∈ Z where ( q, p ) ∈ ∂ Ω × R . � � 2 � � � � c b h n ( q, p ) = ( q,p ) ,k , u n � Husimi function: (96) � Arnd B¨ acker 87 � ⇓ ⇐ ⇒ Σ ⊕

VII More recent results – Poincar´ e Husimi representation Husimi function on the Poincar´ e section P : � � 2 � � � � � 1 � � c b h n ( q, p ) = ( q,p ) ,k n ( s ) u n ( s ) d s . (97) � � 2 πk n � � � � ∂ Ω ([Crespi, Perez, Chang ’93; Tualle,Voros ’95]) Alternative Poincar´ e Husimi representation: � � 2 � � � � � c b ( q,p ) ,k n ( s ) u n ( s ) � ˆ n ( s ) , x ( s ) � d s � � � � 1 ∂ Ω � � h n ( q, p ) = (98) c b 2 k 2 ( q,p ) ,k n ( s ) c b ( q,p ) ,k n ( s ) � ˆ n ( s ) , x ( s ) � d s n ∂ Ω ([Simonotti,Vergini,Saraceno ’97]) Arnd B¨ acker 88 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi functions – examples 1277: 1817: Arnd B¨ acker 89 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi representation – mean behaviour Mean behaviour: For Husimi functions in phase space: � 1 1 H B π vol (Ω) χ Ω ( q ) δ (1 − | p | 2 ) . lim n ( p , q ) = N ( k ) k →∞ k n ≤ k And on the boundary? � 1 h n ( q, p ) → ? H k ( q, p ) := N ( k ) k n ≤ k Arnd B¨ acker 90 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi representation – mean behaviour � 1 Plot of H k ( q, p ) := h n ( q, p ) N ( k ) k n ≤ k Variant 1 Variant 2 a) b) 0.2 0.3 0.2 0.1 0.1 0.0 0.0 L L q q L /2 L /2 1.5 1.5 0.0 0.0 0 0 p p -1.5 -1.5 Arnd B¨ acker 91 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi representation – mean behaviour Analytically we show � � 1 h n ( q, p ) = 2 1 − p 2 + O ( k − 1 / 2 ) , H k ( q, p ) ≡ N ( k ) Aπ k n ≤ k Uniform asymptotics 1.0 k =10 k =30 k =500 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p Arnd B¨ acker 92 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi representation – mean behaviour Numerical comparison of the mean behaviour Section of H k ( q, p ) at q = 3 . 0 0.15 red: numerical result 0.10 blue: uniform 0.05 semiclassics 0.00 p -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Question: Does the ad-hoc definition of the Poincar´ e Husimi functions make sense ? Arnd B¨ acker 93 � ⇓ ⇐ ⇒ Σ ⊕

VII From phase space to the Poincar´ e section Approach: project coherent state in phase space onto boundary q −p x(q) Arnd B¨ acker 94 � ⇓ ⇐ ⇒ Σ ⊕

VII Relation between Husimi functions We show H n ( p , q ) = δ k n (1 − | p | ) 1 h n ( q, p ) 1 − p 2 (1 + O ( k − 1 / 2 � )) , (99) n 4 with � k n � 1 / 2 e − k n (1 −| p | ) 2 . δ k n (1 − | p | ) := (100) π Consequence 1 h n ( q, p ) � � 1 − p 2 � a � ( q, p ) l ( q, p ) d q d p + O ( k − 1 / 2 � ψ n , A ψ n � Ω = ) , (101) n � 4 − 1 ∂ Ω where l ( q, p ) is the length of the orbit segment. Thus: physical interpretation of the Poincar´ e Husimi functions! Arnd B¨ acker 95 � ⇓ ⇐ ⇒ Σ ⊕

VII Quantum ergodicity for Poincar´ e Husimi functions For ergodic systems the quantum ergodicity theorem implies 1 • almost all Husimi functions H n ( p , q ) tend weakly to 2 π vol(Ω) . The relation 1 � � h n ( q, p ) 1 − p 2 � a � ( q, p ) l ( q, p ) d q d p + O ( k − 1 / 2 � ψ n , A ψ n � Ω = ) , (102) n � 4 − 1 ∂ Ω then implies that almost all Poincar´ e Husimi functions � 2 1 − p 2 h n ( q, p ) → (103) π vol(Ω) in the semiclassical limit (in the weak sense). I.e.: Quantum ergodicity theorem for the Poincar´ e Husimi functions Arnd B¨ acker 96 � ⇓ ⇐ ⇒ Σ ⊕

VII Poincar´ e Husimi functions — Summary a) • Mean behaviour of 0.2 Poincar´ e Husimi functions: 0.1 � 0.0 1 − p 2 ∼ L q L /2 1.5 • Relation between 0.0 0 p -1.5 – Husimi functions in phase space and – Poincar´ e Husimi functions. Consequences: – physical interpretation and justification of the previous ad-hoc definitions – quantum ergodicity theorem for the Poincar´ e Husimi functions Arnd B¨ acker 97 � ⇓ ⇐ ⇒ Σ ⊕

VIII More recent results – Time evolution Numerical experiment: Start with coherent state � k � 1 / 2 2 � x − q , ( x − q ) � ] , e i k [ � p , x − q � + i Coh ( p , q ) ,k ( x ) := (104) π where ( p , q ) ∈ R 2 × R 2 denotes the point in phase space around which the coherent state is localized. START Observation: follows classical trajectory for some time Arnd B¨ acker 98 � ⇓ ⇐ ⇒ Σ ⊕

VIII Time evolution Two more examples: What happens for large times? Conjecture: Random wave description START is possible! Arnd B¨ acker 99 � ⇓ ⇐ ⇒ Σ ⊕

VIII Time evolution Conjecture: For chaotic systems the time evolution of an initially localized wavepacket leads to a random state in the limit of large times. One consequence: Gaussian distribution for the components (real and imaginary) of ψ ( q , t ) Or: P ( | ψ | 2 ) = exp( − ψ ) Consider amplitude distribution... Arnd B¨ acker 100 � ⇓ ⇐ ⇒ Σ ⊕