Quantum Chaos on Graphs

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Quantum chaos on graphs

Jon Harrison

Quantum chaos Quantum chaos on graphs Quantum graphs

Trace formula Jon Harrison BGS conjecture

Statistics Baylor University

Dirac op. Graduate seminar – 6th November 07

Jon Harrison Quantum chaos on graphs Outline

Quantum chaos on graphs 1 What is quantum chaos? Jon Harrison

Quantum 2 Everything you always wanted to know about quantum chaos graphs but were afraid to ask. Quantum graphs Trace formula 3 The trace formula. BGS conjecture 4 Statistics The conjecture of Bohigas, Giannoni and Schmidt.

Dirac op. 5 Spectral statistics.

6 The Dirac operator on a graph.

Jon Harrison Quantum chaos on graphs Quantum Mechanics

Quantum chaos on graphs Schr¨odingerequation

Jon Harrison

Quantum ∂Ψ 1 e 2 chaos i~ = −i~∇ − A(x, t) Ψ(x, t) + V (x, t)Ψ(x, t) Quantum ∂t 2m c graphs Trace formula E BGS Time-independent eqn: Ψ(x, t) = ψ(x) exp −i t conjecture ~ Statistics 1 e 2 Dirac op. −i ∇ − A(x) ψ(x) + V (x)ψ(x) = Eψ(x) 2m ~ c Hilbert space L2(x). Energy levels E0 ≤ E1 ≤ E2 ≤ ... . Wave function ψ, |ψ(x)|2dx probability particle is located at x.

Jon Harrison Quantum chaos on graphs Chaos

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula Integrable billiard: regular motion.

BGS conjecture

Statistics

Dirac op.

Chaotic billiard: irregular motion.

Jon Harrison Quantum chaos on graphs Quantum chaos

Quantum chaos on graphs

Jon Harrison Quantum chaos: “Study of Sch¨odinger eqn in classically chaotic systems.” Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Pictures Arnd B¨acker: http://www.physik.tu-dresden.de/∼baecker/

Jon Harrison Quantum chaos on graphs Metric graphs

Quantum chaos on graphs

Jon Harrison v1

Quantum chaos set of vertices V = {vj }

Quantum v2 set of edges E = {ej } v4 graphs v3 Trace formula

BGS conjecture Each edge e corresponds to interval [0, Le ]. Statistics Hilbert space Dirac op. M 2 H := L [0, Le ] e∈E

Jon Harrison Quantum chaos on graphs Laplace op. on a graph

Quantum d2 chaos on Operator on edge − . graphs 2 dxe Jon Harrison Deﬁne a domain of f in H on which the operator is self-adjoint.

Quantum Vertex matching conditions: chaos

Quantum graphs

Trace formula 0 BGS Af(0) + Bf (0) = 0 conjecture

Statistics

Dirac op.

Theorem (Kostrykin & Schrader) The boundary conditions deﬁne a self-adjoint operator if † † rank(A, B) maximal and AB = BA .

Jon Harrison Quantum chaos on graphs Vertex scattering matrix

Quantum chaos on 2 graphs d 2 − 2 ψe (xe ) = k ψe (xe ) Jon Harrison dxe

Quantum Plane-wave solutions chaos ikxe −ikxe Quantum ψe (xe , k) = ce e + de e graphs

Trace formula From matching conditions at a vertex

BGS conjecture A(c + d) + ikB(c − d) = 0 . (1) Statistics

Dirac op. Vertex scattering matrix, d = Σ c.

−1 Σ = −(A − ikB) (A + ikB) † AB self-adjoint implies Σ unitary.

Jon Harrison Quantum chaos on graphs Example: Neumann matching conditions

X Quantum f is continuous at vertex v and f 0(0) = 0. chaos on e graphs e∼v

Jon Harrison  1 −1 0 0 ...  0 1 −1 0 ... 0 0 ... 0 ! Quantum . . . chaos A =  .. ..  B = . . .  . .  0 0 ... 0 Quantum 0 ... 0 1 −1 1 1 ... 1 graphs 0 ... 0 0 0 Trace formula Vertex scattering matrix, BGS conjecture  2 2  d −1 d Statistics v v Σ = . Dirac op.  ..  2 2 −1 dv dv Neumann transition amplitude σ = 2 − δ . ef dv ef From now on we assume Σ independent of k.

Jon Harrison Quantum chaos on graphs Scattering matrix

Quantum chaos on graphs

Jon Harrison (v) ikL(vy) (uv)(wy)(k) := δvw σ e dim( (k)) = 2|E| Quantum S (uv)(vy) S chaos

Quantum Eigenfunctions ψ correspond to vectors c of all plane-wave graphs coeﬃcients where Trace formula S(k)c = c BGS conjecture Statistics Quantization condition Dirac op. Eigenvalues kn are solutions of

|I − S(k)| = 0

Jon Harrison Quantum chaos on graphs Trace formula – Roth, Kottos & Smilansky

Quantum chaos on graphs Deﬁne ζ(k) := |I − S(k)| so ζ(kn) = 0. iφj (k) Jon Harrison Let e be an eigenvalue of S(k).

Quantum 2|E| 2|E| chaos Y iφ (k) 2|E| 1 Y φj ζ(k) = (1 − e j ) = 2 |S(k)| 2 sin Quantum 2 graphs j=1 j=1 Trace formula 1 BGS − 2 conjecture ζ(k)|S(k)| is a real fn. with zeros at kn.

Statistics ∞ Dirac op. X d(k) = δ(k − kn) n=1 1 d − 1 = − lim Im log ζ(k + i) |S(k + i)| 2 π →0 dk

Jon Harrison Quantum chaos on graphs Weyl term

Quantum chaos on graphs

Jon Harrison

1 d − 1 Quantum dWeyl(k) := − lim Im log(|S(k + i)| 2 ) chaos π →0 dk Quantum 1 d graphs = Im log(e−ikL) Trace formula 2π dk L BGS = conjecture 2π Statistics

Dirac op. X Total length of graph L := Le . e∈E L Mean separation of eigenvalues . 2π

Jon Harrison Quantum chaos on graphs Oscillating part of spectral density

Quantum chaos on ∞ graphs X 1 n log(|I2B − S(k)|) = − tr S (k) Jon Harrison n n=1 Quantum X chaos n ikLe1 ikLe2 ikLbn tr S (k) = e σe1e2 e σe2e3 ... e σene1 Quantum e1,...,en graphs n Trace formula If tr S (k) 6= 0 then (e1e2 ... en) is a periodic orbit p.

BGS iπµp conjecture σe1e2 σe2e3 . . . σene1 := Ape Statistics Putting it together Dirac op. n X iπµp ikLp tr S (k) = n Ape e

p∈Pn

1 d 1 X Lp iπµp − lim Im log ζ(k + i) = Ape cos(kLp) π →0 dk π r p p

Jon Harrison Quantum chaos on graphs Famous cousins in the trace family

Quantum chaos on Quantum graph graphs

Jon Harrison L 1 X Lp iπµp d(k) = + Ape cos(kLp) Quantum 2π π rp chaos p

Quantum graphs Gutzwiller’s trace formula – chaotic Hamiltonian system.

Trace formula 2 X 1 BGS d(E) ∼ d(E) + A cos (S + µ ) conjecture p p p ~ p ~ Statistics Dirac op. Selberg’s trace formula

∞ ∞ ∞ X Area(M) Z X X Lphˆ(nLp) h(ρj ) = h(ρ) tanh(πρ)ρdρ + 4π 2 sinh(nL /2) j=0 −∞ p n=1 p

ρj evalue of Laplace-Beltrami op on compact hyperbolic surface M.

Jon Harrison Quantum chaos on graphs Poisson summation formula

Quantum chaos on graphs ∞ ∞ Jon Harrison X X h(j) = hˆ(2πp) Quantum p=−∞ chaos j=−∞ Quantum This is the trace formula of a graph with a single edge L = 2π. graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Neumann matching condition at the vertex σee = 1, σee = 0.

Jon Harrison Quantum chaos on graphs The zeta function

Quantum chaos on graphs ∞ −1 X 1 Y 1 Jon Harrison ζ(s) = = 1 − ns ps Quantum n=1 p prime chaos

Quantum graphs Trivial zeros s = −2, −4,... . Trace formula Other zeros in critical strip BGS conjecture 0 < Re(s) < 1. Statistics Dirac op. Riemann Hypothesis: All complex zeros lie on 1 the line Re(s) = . 2

Jon Harrison Quantum chaos on graphs An even more famous cousin

Quantum chaos on graphs

Jon Harrison

Quantum Riemann-Weil explicit formula chaos X Quantum h(γ ) = h(i/2) + h(−i/2) − hˆ(0) log π graphs j j Trace formula ∞ 0 ∞ BGS 1 Z Γ 1 1 X Λ(n) conjecture + h(ρ) + iρ dρ − 2 √ hˆ(log n) 2π Γ 4 2 n Statistics −∞ n=1 Dirac op. where 1/2 + iγj non-trival zero of ζ.

Jon Harrison Quantum chaos on graphs Classical dynamics on graphs

Quantum chaos on graphs

Jon Harrison Probabilistic dynamics Quantum chaos M matrix of transition probabilities, Quantum graphs

Trace formula ρn+1 = Mρn .

BGS 2 conjecture M(uv)(vw) = |S(uv)(vw)|

Statistics

Dirac op. As S is unitary M is doubly stochastic - rows and columns sum to 1.

Jon Harrison Quantum chaos on graphs Chaotic properties

Quantum chaos on graphs For a connected graph the Markov chain is ergodic,

Jon Harrison q (M )ef > 0 for some q . Quantum chaos M has an eigenvalue 1, if there are no other eigenvalues on the Quantum graphs unit circle the graph is mixing: Trace formula BGS n 1 T n 1 conjecture lim M ρ = (1,..., 1) or lim (M )ef = n→∞ 2|E| n→∞ 2|E| Statistics

Dirac op. Note: for Pn the set of periodic orbits of length n. X lim tr Mn = lim n A2 = 1 n→∞ n→∞ p p∈Pn

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Jon Harrison Quantum chaos on graphs The Gaussian unitary ensemble

Quantum chaos on graphs

Jon Harrison

Quantum T chaos Ensemble of N × N Hermitian matrices, H = H. Quantum graphs Independent matrix elements uncorrelated. Trace formula Probability density P(H) invariant under unitary BGS conjecture transformations. −A tr(H2) Statistics Uniquely determines P(H) = Ce . Dirac op.

Jon Harrison Quantum chaos on graphs BGS

Quantum chaos on graphs

Jon Harrison

Quantum Conjecture – Bohigas, Giannoni, Schmit chaos

Quantum In the semiclassical limit energy level statistics of a quantum graphs system whose classical analogue is chaotic correspond to Trace formula eigenvalue statistics of random matrix ensembles. BGS conjecture Statistics Gaussian Unitary Ensemble (GUE) no time-reversal symmetry Dirac op. Gaussian Orthogonal Ensemble (GOE) time-reversal symm., T 2 = I Gaussian Symplectic Ensemble (GSE) time-reversal symm., T 2 = −I

Jon Harrison Quantum chaos on graphs Nearest neighbor spacing statistics

Quantum chaos on graphs s1 s2 s3 s4 ...

Jon Harrison

Quantum chaos

Quantum graphs Trace formula Deﬁnition (Level spacing distribution) BGS conjecture P(s) probability density of spacings between consecutive Statistics eigenvalues. Dirac op. Integrated spacing distribution Z s I (s) = P(t)dt 0

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum Gaussian unitary ensemble of random matrices chaos 2 Quantum 32s −4s2 graphs π PGUE(s) ≈ 2 e . (2) Trace formula π

BGS conjecture Poisson distribution for uniform random numbers

Statistics −s Dirac op. PPoisson(s) = e . (3)

Jon Harrison Quantum chaos on graphs Spacings of evalues of large random matrix

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

PGUE(s) Jon Harrison Quantum chaos on graphs Spacing distributions for quantum graph no time-reversal symmetry, 25275 levels.

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Jon Harrison Quantum chaos on graphs Spacing distribution of the zeros of zeta

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

PGUE(s) Jon Harrison Quantum chaos on graphs Hilbert-Polya conjecture

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics

Dirac op.

Jon Harrison Quantum chaos on graphs Dirac operator on a graph – J.H. & J. Bolte

Quantum chaos on graphs Dirac eqn. in 1d Jon Harrison ∂ ∂ 2 Quantum i~ Ψ(x, t) = −i~c α + mc β Ψ(x, t) (4) chaos ∂t ∂x Quantum graphs 0 −i 1 0 α = i 0 β = 0 −1 Trace formula

BGS conjecture Trace formula Statistics

Dirac op. 2L 1 X Lp d(k) = + A eiπµp tr(u ) cos(kL ) (5) π π r p p p p p

where up ∈ G ⊆ SU(2).

Jon Harrison Quantum chaos on graphs Spacing distributions for the Dirac op. time-reversal symmetric T 2 = −I.

Quantum chaos on graphs

Jon Harrison

Quantum chaos

Quantum graphs

Trace formula

BGS conjecture

Statistics Dirac op. To obtain GSE statistics G irreducible quaternionic representation.

Quaternionic repn. – equivalent to complex conjugate repn. but not equivalent to real repn.

Jon Harrison Quantum chaos on graphs Final remarks

Quantum chaos on graphs

Jon Harrison Quantum graphs are a simple model for complex spectral Quantum chaos problems. Quantum In the semi-classical limit spectral statistics preﬁgure graphs

Trace formula classical chaos. BGS Trace formulae are great. conjecture Statistics To do: Dirac op. Prove the B-G-S conjecture on a graph. Prove quantum ergodicity of eigenfunctions on a graph.

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum S. DeBievre, “Quantum chaos: a brief ﬁrst visit,” chaos Contemporary Mathematics 289, 161-218 (2001). Quantum graphs http://www.ma.utexas.edu/mp arc-bin/mpa?yn=01-207 Trace formula S. Gnutzmann & U. Smilansky, “Quantum Graphs: BGS conjecture Applications to Quantum Chaos and Universal Spectral Statistics Statistics,” Advances in Physics, 55, 527 (2006). Dirac op. arXiv:nlin.CD/0605028

Jon Harrison Quantum chaos on graphs Quantum chaos on graphs

Jon Harrison

Quantum “He had brought a large map representing the sea, chaos Without the least vestige of land: Quantum graphs And the crew were much pleased when they found it to be Trace formula A map they could all understand.” BGS conjecture

Statistics

Dirac op. Lewis Carroll – The Hunting of the Snark.

Jon Harrison Quantum chaos on graphs