Quantum chaos on graphs

Transcription

1 Baylor University Graduate seminar 6th November 07

2 Outline 1 What is quantum? 2 Everything you always wanted to know about quantum but were afraid to ask. 3 The trace formula. 4 The of Bohigas, Giannoni and Schmidt. 5 Spectral statistics. 6 The Dirac operator on a graph.

3 Mechanics Schrödinger equation i Ψ t = 1 ( i e ) 2 2m c A(x, t) Ψ(x, t) + V (x, t)ψ(x, t) Time-independent eqn: Ψ(x, t) = ψ(x) exp ( i E ) t 1 ( i e ) 2 2m c A(x) ψ(x) + V (x)ψ(x) = Eψ(x) Hilbert space L 2 (x). Energy levels E 0 E 1 E Wave function ψ, ψ(x) 2 dx probability particle is located at x.

4 Chaos Integrable billiard: regular motion. Chaotic billiard: irregular motion.

5 : Study of Schödinger eqn in classically chaotic systems. Pictures Arnd Bäcker: baecker/

6 Metric v 1 v 3 v 4 v 2 set of vertices V = {v j } set of edges E = {e j } Each edge e corresponds to interval [0, L e ]. Hilbert space H := L 2( [0, L e ] ) e E

7 Laplace op. on a graph Operator on edge d2 dx 2 e. Define a domain of f in H on which the operator is self-adjoint. Vertex matching conditions: Af(0) + Bf (0) = 0 Theorem (Kostrykin & Schrader) The boundary conditions define a self-adjoint operator if rank(a, B) maximal and AB = BA.

8 Vertex scattering matrix d2 dxe 2 ψ e (x e ) = k 2 ψ e (x e ) Plane-wave solutions ψ e (x e, k) = c e e ikxe + d e e ikxe From matching conditions at a vertex A(c + d) + ikb(c d) = 0. (1) Vertex scattering matrix, d = Σ c. Σ = (A ikb) 1 (A + ikb) AB self-adjoint implies Σ unitary.

9 Example: Neumann matching conditions f is continuous at vertex v and e v A = Vertex scattering matrix, Σ = f e (0) = 0. B = 2 dv 1 2 dv... 2 dv 2 dv 1 ( ) Neumann transition amplitude σ ef = 2 d v δ ef. From now on we assume Σ independent of k.

10 Scattering matrix S (uv)(wy) (k) := δ vw σ (v) (uv)(vy) eikl (vy) dim(s(k)) = 2 E Eigenfunctions ψ correspond to vectors c of all plane-wave coefficients where S(k)c = c Quantization condition Eigenvalues k n are solutions of I S(k) = 0

11 Roth, Kottos & Smilansky Define ζ(k) := I S(k) so ζ(k n ) = 0. Let e iφ j (k) be an eigenvalue of S(k). 2 E 2 E ( ) ζ(k) = (1 e iφ j (k) ) = 2 2 E S(k) 1 φj 2 sin 2 j=1 ζ(k) S(k) 1 2 is a real fn. with zeros at k n. d(k) = δ(k k n ) n=1 j=1 = 1 π lim Im d ( ) ɛ 0 dk log ζ(k + iɛ) S(k + iɛ) 1 2

12 Weyl term d Weyl (k) := 1 π lim ɛ 0 Im d dk log( S(k + iɛ) 1 2 ) = 1 2π Im d dk log(e ikl ) = L 2π Total length of graph L := e E L e. Mean separation of eigenvalues L 2π.

13 Oscillating part of spectral density tr S n (k) = log( I 2B S(k) ) = n=1 1 n tr S n (k) e 1,...,e n e ikle1 σe1 e 2 e ikle2 σe2 e 3... e ikl bn σene 1 If tr S n (k) 0 then (e 1 e 2... e n ) is a periodic orbit p. Putting it together σ e1 e 2 σ e2 e 3... σ ene 1 := A p e iπµp tr S n (k) = n p P n A p e iπµp e iklp 1 π lim Im d ɛ 0 dk log ζ(k + iɛ) = 1 L p A p e iπµp cos(kl p ) π r p p

14 Famous cousins in the trace family graph d(k) = L 2π + 1 L p A p e iπµp cos(kl p ) π r p p Gutzwiller s trace formula chaotic Hamiltonian system. d(e) d(e) + 2 A p cos 1 (S p + µ p ) Selberg s trace formula p j=0 h(ρ j ) = Area(M) 4π h(ρ) tanh(πρ)ρdρ + p n=1 L p ĥ(nl p) 2 sinh(nl p/2) ρ j evalue of Laplace-Beltrami op on compact hyperbolic surface M.

15 Poisson summation formula h(j) = ĥ(2πp) j= p= This is the trace formula of a graph with a single edge L = 2π. Neumann matching condition at the vertex σ ee = 1, σ ee = 0.

16 The zeta function ζ(s) = n=1 1 n s = p prime Trivial zeros s = 2, 4,.... Other zeros in critical strip 0 < Re(s) < 1. Riemann Hypothesis: All complex zeros lie on the line Re(s) = 1 2. (1 1p s ) 1

17 An even more famous cousin Riemann-Weil explicit formula h(γ j ) = h(i/2) + h( i/2) ĥ(0) log π j + 1 ( h(ρ) Γ 1 2π Γ ) 2 iρ dρ 2 where 1/2 + iγ j non-trival zero of ζ. n=1 Λ(n) n ĥ(log n)

18 Classical dynamics on Probabilistic dynamics M matrix of transition probabilities, ρ n+1 = Mρ n. M (uv)(vw) = S (uv)(vw) 2 As S is unitary M is doubly stochastic - rows and columns sum to 1.

19 Chaotic properties For a connected graph the Markov chain is ergodic, (M q ) ef > 0 for some q. M has an eigenvalue 1, if there are no other eigenvalues on the unit circle the graph is mixing: lim n Mn ρ = 1 2 E (1,..., 1)T or lim n (Mn ) ef = 1 2 E Note: for P n the set of periodic orbits of length n. lim tr n Mn = lim n A 2 n p = 1 p P n

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22 The Gaussian unitary ensemble Ensemble of N N Hermitian matrices, H T = H. Independent matrix elements uncorrelated. Probability density P(H) invariant under unitary transformations. Uniquely determines P(H) = Ce A tr(h2).

23 Conjecture Bohigas, Giannoni, Schmit In the semiclassical limit energy level statistics of a quantum system whose classical analogue is chaotic correspond to eigenvalue statistics of random matrix ensembles. Gaussian Unitary Ensemble (GUE) Gaussian Orthogonal Ensemble (GOE) Gaussian Symplectic Ensemble (GSE) no time-reversal symmetry time-reversal symm., T 2 = I time-reversal symm., T 2 = I

24 Nearest neighbor spacing statistics s 1 s 2 s 3 s 4... Definition (Level spacing distribution) P(s) probability density of spacings between consecutive eigenvalues. Integrated spacing distribution I (s) = s 0 P(t)dt

25 Gaussian unitary ensemble of random matrices P GUE (s) 32s2 π 2 e 4s 2 π. (2) Poisson distribution for uniform random numbers P Poisson (s) = e s. (3)

26 Spacings of evalues of large random matrix s PGUE (s)

27 Spacing distributions for quantum graph no time-reversal symmetry, levels.

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29 Spacing distribution of the zeros of zeta s PGUE (s)

30 Hilbert-Polya

31 Dirac operator on a graph J.H. & J. Bolte Dirac eqn. in 1d i t Ψ(x, t) = ( i c α x + mc2 β α = ( ) 0 i i 0 d(k) = 2L π + 1 π where u p G SU(2). p β = ( ) ) Ψ(x, t) (4) L p r p A p e iπµp tr(u p ) cos(kl p ) (5)

32 Spacing distributions for the time-reversal symmetric T 2 = I. To obtain GSE statistics G irreducible quaternionic representation. Quaternionic repn. equivalent to complex conjugate repn. but not equivalent to real repn.

33 Final remarks To do: are a simple model for complex spectral problems. In the semi-classical limit spectral statistics prefigure classical. e are great. Prove the B-G-S on a graph. Prove quantum ergodicity of eigenfunctions on a graph.

34 S. DeBievre, : a brief first visit, Contemporary Mathematics 289, (2001). arc-bin/mpa?yn= S. Gnutzmann & U. Smilansky, Graphs: Applications to Chaos and Universal Spectral, Advances in Physics, 55, 527 (2006). arxiv:nlin.cd/

35 He had brought a large map representing the sea, Without the least vestige of land: And the crew were much pleased when they found it to be A map they could all understand. Lewis Carroll The Hunting of the Snark.