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The Transfer Operator Approach to ``Quantum Chaos

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The Transfer Operator Approach to ``Quantum Chaos Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Front page The transfer operator approach to “quantum chaos” Classical mechanics and the Laplace-Beltrami operator on PSL (Z) H 2 \ Tobias M¨uhlenbruch Joint work with D. Mayer and F. Str¨omberg Institut f¨ur Mathematik TU Clausthal [email protected] 22 January 2009, Fernuniversit¨at in Hagen Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Outline of the presentation 1 Modular surface 2 Spectral theory 3 Geodesics and coding 4 Transfer operator 5 Connection to spectral theory 6 Conclusions Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z The full modular group 2( ) Let PSL2(Z) be the full modular group PSL (Z)= S, T (ST )3 =1 = SL (Z) mod 1 2 h | i 2 {± } 0 1 1 1 1 0 with S = − , T = and 1 = . 1 0 0 1 0 1 a b az+b if z C, M¨obius transformations: z = cz+d ∈ c d a if z = . c ∞ Three orbits: the upper halfplane H = x + iy; y > 0 , the 1 { } projective real line PR and the lower half plane. z , z are PSL (Z)-equivalent if M PSL (Z) with 1 2 2 ∃ ∈ 2 M z1 = z2. The full modular group PSL2(Z) is generated by 1 translation T : z z + 1 and inversion S : z − . 7→ 7→ z (Closed) fundamental domain = z H; z 1, Re(z) 1 . F ∈ | |≥ | |≤ 2 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z H The modular surface 2( )\ The upper half plane H = x + iy; y > 0 can be viewed as a hyperbolic plane with constant{ negative} curvature 1. − d d 2 dx2+dy 2 Line element s s = y 2 d d dxdy Volume element A A = y 2 The M¨obius transformations are compatible with the hyperbolic metric. This way, it makes sense to speak of the modular surface PSL (Z) H. 2 \ Gluing along the identified edges, we obtain a realization of the modularF surface PSL (Z) H, a non-compact, finite surface with one 2 \ cusp at z i and two conic singularities at z = i and z = 1+i√3 . → ∞ 2 The hyperbolic plane H is also called the Poincar´ehalf plane. PSL (Z) H is arithmetic ( connection to number theory). 2 \ Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Hecke triangle groups Possible extension of PSL2(Z): Hecke triangle groups Gq G = S, T (ST )q =1 with q h q| q i 1 λ T = q , λ = 2cos π/q and q =3, 4, 5,.... 0 1 q Example: PSL2(Z)= G3. M¨obius transformation extends to Gq. Fundamental domain Re λq q = z H; z 1, (z) 2 . F n ∈ | |≥ | |≤ o Fundamental domain forms a (2, q, ) π π∞ hyperbolic triangle with angles 2 , q and 0. G H is finite and non-compact. q\ G H is non-arithmetic for q =3, 4, 6. q\ 6 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Maass cusp forms Maass cusp form u u : H C real-analytic function, → ∆u = s(1 s) u with ∆ = y 2 ∂2 + ∂2 , − − x y u(M z)= u(z) for all M PSL (Z), ∈ 2 u(x + iy)= y C as y for all C R. O → ∞ ∈ Maass cusp form at s = 1 + i13.7797 .... ∆ admits self-adjoined extension in 2 L2 PSL (Z) H . 2 \ s is called spectral parameter. s(1 s) > 1 , i.e., s 1 + iR⋆. − 4 ∈ 2 Discrete spectrum, eigenvalues have finite multiplicity, It is assumed that eigenvalues have multiplicity 1. Precise location of eigenvalues is unknown. 2πinx Maass cusp form at u(x + iy)= √y 0=n Z an Ks 1 (2π n y) e 6 ∈ − 2 | | s = 1 + i9.533 .... P 2 Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The Laplace-Beltrami operator L2 PSL (Z) H are realized by functions h satisfying 2 \ h : H C measurable, → h(M z)= f (z) for all M PSL (Z) and ∈ 2 2 2 h(x, iy) y − dxdy < . | | ∞ RF Some properties of ∆ The Laplace-Beltrami operator on L2(M) is the self-adjoined extension of ∆. Spectrum of ∆ is discrete. Area(H) Eigenvalues λ = s(1 − s) obey Weyl’s law: ♯{λ ≤ Λ}∼ 4π . Conjectures about Eigenvalue statistics. Quantum unique ergodicity. Arithmetic properties: Hecke operators, associated L-series satisfy a GRH. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Hurwitz continued fractions Definition (Hurwitz continued fractions (CF)) ⋆ We identify a sequence of integers, a0 ∈ Z, and a1, a2,... ∈ Z with a point a0 a1 a2 −1 x = T ST ST · · · 0= a0 + −1 =: [a0; a1, a2,...] a1 + −1 a2+ ... and say that it is a non-regular (formal) CF, [a0; a1, a2,...] in general. regular CF, [a0; a1, a2,...], if it does not contain “forbidden blocks”: no ±1 appear and if ai = ±2 then ai+1 ≶ 0. π = [3; 7, 16, 294, 3, 4, 5, 15,...] and e = [3;4, 2, 5, 2, 7, 2, 9,...] − − − − Equivalent points x and y are equivalent :⇔ there exist a g ∈ PSL2(Z) such that gx = y ⇔ the CF of x and y have the same tail or the CF of x and y have tail [ 3] and [ −3 ]. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Associated dynamical system The generating map f (x) Z denotes the nearest integer of x, i.e. x (x) 1 . ∈ | − |≤ 2 For I = 1 , 1 the generating map for the CF of x is − 2 2 1 1 f : I I ; x − − . → 7→ x − x The coefficients a0, a1,... computed by 1 a0 = (x) and xn+1 = f (xn)= − an+1 xn − satisfy x =[a0; a1,...] and the CF is regular. Natural extension of f The natural extension of f is 1 Ω Ω; (x, y) f (x), − → 7→ y + a1 with x = [0; a1,...]. Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions PSL Z 2( ) and geodesics Recall PSL (Z)= S, T . 2 h i We denote geodesics γ on H by its base points: γ = (γ ,γ+). − In the diagram we illustrate the closed geodesic 1 γ = ([0; 3, 4], [0; 4, 3]− ). − − − − Theorem Each geodesic γ′ is PSL2(Z)-equivalent with a geodesic 1 γ = (γ ,γ+) satisfying (γ ,γ+− ) Ω. − − ∈ If γ′ is closed then γ = [0; a1,..., an] is regular and 1 − γ+− = [0; an,..., a1] 1 1 If γ and υ satisfy (γ ,γ+− ), (υ ,υ+− ) Ω and no base point is − − ∈ PSL 3 √5 2(Z)-equivalent with [0; 3] = −2 then γ = υ. Nearest λ-multiple continued fractions Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The Ising model and transfer matrices Ernst Ising (1900 – 1998) discussed an 1 − d lattice spin model. Ising-Model N Config. space S = {±1} ; left-shift τ : S → S, (τξ)i := ξi+1. ∞ ∞ Total energy: E = −J i=1 ξi ξi+1 + B i=1 ξi with spin interaction J and magnetic field interaction B.) P P Ernst Ising ≈ 1925. Partition function Zm(A, s) m−1 τ k ξ −s k=0 A( ) 1 Zm(A, s)= e P with s = Temp. and A(ξ)= J ξ0ξ1 + Bξ0. ξ∈S; m−Xperiodic −1 free energy = limm→∞ m log Zm(A, s). Ising rewrote Zm as −s(Jξi ξj −Bξi ) Zm(A, s)= e(ξ1,ξ2) · · · e(ξm,ξ1) with e(ξi ,ξj )= e . ξ1,...,ξXm ∈{±1} Transfer matrix Ls e(+1, +1) e(+1, −1) L := satisfies Z (A, s) = trace Lm . s e(−1, +1) e(−1, −1) m s „ « ` ´ Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions General form of a transfer operator General form of a transfer operator Given a set Λ and maps f : Λ Λ and g : Λ C, a transfer operator acting on→ functions h :→ Λ C is defined by L → h (x)= g(y) h(y) L y Xf −1(x) ∈ Remarks: 1 Usually, take g = J − if the Jacobian J of f exists. | | 1 of the form h(x)= −1 f ′(y) − h(y) is also known as a L L y f (x) Perron-Frobenius OperatorP .∈ Relation of to the dynamical zeta-function: more L 1 ζ(z)= . detc (1 z ) − L Front Modular surface Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions The transfer operator for continued fractions Recall respectively define: 0 −1 1 1 PSL S = 1 0 and T = 0 1 ∈ 2(Z), ` a b ´ ` ´2 −s az+b h 2s c d (z) := (cz + d) h cz+d and `f :˛ −` 1 ,´´1 → −`1 , 1 ; x ´7→ −1 `(mod´ 1) = T −nS x for some n ∈ Z. ˛ 2 2 2 2 x ˆ ˜ ˆ ˜ The associated transfer operator is formally given by s h(x) := −1 f ′(y) − h(y) (x [ 1/2, 1/2]) Ls y f (x) ∈ − P ∈ ω Banach space (with sup-norm) V := C(D) C (D◦), D = z; z 1 . ∩ { | |≤ } Transfer operator for continued fractions The operator : : V V V V , Re(s) > 1, is defined as Ls × → × ∞ h ST n + ∞ h ST n ~ n=3 1 2s n=2 2 2s − s h(z)= n n L P∞ h1 ST + P∞ h2 ST − n=2 2s n=3 2s P P h for ~h = 1 V V .
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