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~r − 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) = (x) exp −i t conjecture ~ Statistics 1 e 2 Dirac op. −i r − A(x) (x) + V (x) (x) = E (x) 2m ~ c Hilbert space L2(x). Energy levels E0 ≤ E1 ≤ E2 ≤ ::: . Wave function , j (x)j2dx 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 = fvj g Quantum v2 set of edges E = fej g 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 ] e2E 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 y y rank(A; B) maximal and AB = BA . Jon Harrison Quantum chaos on graphs Vertex scattering matrix Quantum chaos on 2 graphs d 2 − 2 e (xe ) = k e (xe ) Jon Harrison dxe Quantum Plane-wave solutions chaos ikxe −ikxe Quantum e (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) y 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 0 1 −1 0 0 ::: 1 0 1 −1 0 ::: 0 0 ::: 0 ! Quantum . chaos A = B .. .. C B = . @ . A 0 0 ::: 0 Quantum 0 ::: 0 1 −1 1 1 ::: 1 graphs 0 ::: 0 0 0 Trace formula Vertex scattering matrix, BGS conjecture 0 2 2 1 d −1 d Statistics v v Σ = . Dirac op. @ .. A 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)) = 2jEj Quantum S (uv)(vy) S chaos Quantum Eigenfunctions 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 jI − S(k)j = 0 Jon Harrison Quantum chaos on graphs Trace formula { Roth, Kottos & Smilansky Quantum chaos on graphs Define ζ(k) := jI − S(k)j so ζ(kn) = 0. iφj (k) Jon Harrison Let e be an eigenvalue of S(k). Quantum 2jEj 2jEj chaos Y iφ (k) 2jEj 1 Y φj ζ(k) = (1 − e j ) = 2 jS(k)j 2 sin Quantum 2 graphs j=1 j=1 Trace formula 1 BGS − 2 conjecture ζ(k)jS(k)j is a real fn. with zeros at kn. Statistics 1 Dirac op. X d(k) = δ(k − kn) n=1 1 d − 1 = − lim Im log ζ(k + i) jS(k + i)j 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(jS(k + i)j 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 . e2E L Mean separation of eigenvalues . 2π Jon Harrison Quantum chaos on graphs Oscillating part of spectral density Quantum chaos on 1 graphs X 1 n log(jI2B − S(k)j) = − 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 p2Pn 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 1 1 1 X Area(M) Z X X Lph^(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 1 1 Jon Harrison X X h(j) = h^(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 −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) − h^(0) log π graphs j j Trace formula 1 0 1 BGS 1 Z Γ 1 1 X Λ(n) conjecture + h(ρ) + iρ dρ − 2 p h^(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) = jS(uv)(vw)j 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!1 2jEj n!1 2jEj Statistics Dirac op. Note: for Pn the set of periodic orbits of length n. X lim tr Mn = lim n A2 = 1 n!1 n!1 p p2Pn 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.