The Transfer Operator Approach to ``Quantum Chaos

Front <ul style="display: flex;"><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Front page The transfer operator approach to “quantum chaos” Classical mechanics and the Laplace-Beltrami operator on PSL2(Z)\H Tobias Mu¨hlenbruch Joint work with <a href="/goto?url=http://www.dynamik.tu-clausthal.de/institute/" target="_blank">D. Mayer </a>and <a href="/goto?url=http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg.php" target="_blank">F. Str</a><a href="/goto?url=http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg.php" target="_blank">¨</a><a href="/goto?url=http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/stroemberg.php" target="_blank">omberg </a><a href="/goto?url=http://www.math.tu-clausthal.de/" target="_blank">Institut f</a><a href="/goto?url=http://www.math.tu-clausthal.de/" target="_blank">u</a><a href="/goto?url=http://www.math.tu-clausthal.de/" target="_blank">¨r Mathematik </a> <a href="/goto?url=http://www.tu-clausthal-de" target="_blank">TU Clausthal </a><a href="mailto:[email protected]" target="_blank">[email protected] </a>22 January 2009, <a href="/goto?url=http://www.fernuni-hagen.de/" target="_blank">Fernuniversit</a><a href="/goto?url=http://www.fernuni-hagen.de/" target="_blank">¨</a><a href="/goto?url=http://www.fernuni-hagen.de/" target="_blank">at in Hagen </a>Front <ul style="display: flex;"><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Outline of the presentation 1Modular surface 23456Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions Front Modular surface <ul style="display: flex;"><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>The full modular group PSL (Z) 2Let PSL2(Z) be the full modular group PSL2(Z) = hS, T|(ST)3 = 1i = SL2(Z) mod { 1} <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>01<ul style="display: flex;"><li style="flex:1">−1 </li><li style="flex:1">1</li></ul>0 11 10 01<ul style="display: flex;"><li style="flex:1">with S = </li><li style="flex:1">, T = </li><li style="flex:1">and 1 = </li><li style="flex:1">.</li></ul>0<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul>az+b cz+d <ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">b</li></ul>if z ∈ C, <ul style="display: flex;"><li style="flex:1">Möbius transformations: </li><li style="flex:1">z = </li></ul>a<ul style="display: flex;"><li style="flex:1">c</li><li style="flex:1">d</li></ul>if z = ∞. cThree orbits: the upper halfplane H = {x + iy; y > 0}, the projective real line P1R and the lower half plane. z1, z2 are PSL2(Z)-equivalent if ∃M ∈ PSL2(Z) with M z1 = z2. The full modular group PSL2(Z) is generated by−1 <ul style="display: flex;"><li style="flex:1">translation T : z → z + 1 and inversion S : z → </li><li style="flex:1">.</li></ul>zꢃ12(Closed) fundamental domain F = z ∈ H; |z| ≥ 1, |Re(z)| ≤ .Front Modular surface <ul style="display: flex;"><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>The modular surface PSL (Z)\H 2The upper half plane H = {x + iy; y > 0} can be viewed as a hyperbolic plane with constant negative curvature −1. dx2+dy2 Line element ds ds2 = y2 dxdy Volume element dA dA = y2 The Möbius transformations are compatible with the hyperbolic metric. This way, it makes sense to speak of the modular surface PSL2(Z)\H. Gluing F along the identified edges, we obtain a realization of the modular surface PSL2(Z)\H, a non-compact, finite surface with √one 1+i 2 3cusp at z → i∞ and two conic singularities at z = i and z = The hyperbolic plane H is also called the Poincaré half plane. PSL2(Z)\H is arithmetic ( connection to number theory). .Front Modular surface <ul style="display: flex;"><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Hecke triangle groups Possible extension of PSL2(Z): Hecke triangle groups Gq Gq = hS, Tq|(STq)q = 1i with , λq = 2 cos π/q and q = 3, 4, 5, . . .. <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul>10λq 1T = Example: PSL2(Z) = G3. Möbius transformation extends to Gq. Fundamental domain <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">o</li></ul>λq Fq = z ∈ H; |z| ≥ 1, |Re(z)| ≤ .2Fundamental domain forms a (2, q, ∞) π2πq<ul style="display: flex;"><li style="flex:1">hyperbolic triangle with angles </li><li style="flex:1">,</li><li style="flex:1">and 0. </li></ul>Gq\H is finite and non-compact. Gq\H is non-arithmetic for q = 3, 4, 6. <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li></ul>Spectral theory <ul style="display: flex;"><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Maass cusp forms Maass cusp form u u : H → C real-analytic function, <ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul><ul style="display: flex;"><li style="flex:1">∆u = s(1 − s) u with ∆ = −y2 ∂x2 + ∂y2 </li><li style="flex:1">,</li></ul>u(M z) = u(z) for all M ∈ PSL2(Z), <ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul>u(x + iy) = O yC as y → ∞ for all C ∈ R. Maass cusp form at 1s = + i13.7797 . . .. ∆ admits self-adjoined extension in L2 PSL2(Z)\H . 2<ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul>s is called spectral parameter. s(1 − s) > 14 , i.e., s ∈ + iR⋆. 12Discrete spectrum, eigenvalues have finite multiplicity, It is assumed that eigenvalues have multiplicity 1. Precise location of eigenvalues is unknown. P√(2π |n| y) e2πinx Maass cusp form at u(x + iy) = <ul style="display: flex;"><li style="flex:1">y</li><li style="flex:1">an K </li></ul>1s− 2 0=n∈Z 12s = + i9.533 . . .. <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li></ul>Spectral theory <ul style="display: flex;"><li style="flex:1">Geodesics and coding </li><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>The Laplace-Beltrami operator <ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul>L2 PSL2(Z)\H are realized by functions h satisfying h: H → C measurable, h(M z) = f (z) for all M ∈ PSL2(Z) and R2F |h(x, iy)| y−2dxdy < ∞. Some properties of ∆ The Laplace-Beltrami operator on L2(M) is the self-adjoined extension of ∆. Spectrum of ∆ is discrete. Area(H) 4π Eigenvalues λ = s(1 − s) obey Weyl’s law: ♯{λ ≤ Λ} ∼ Conjectures about Eigenvalue statistics. Quantum unique ergodicity. . Arithmetic properties: Hecke operators, associated L-series satisfy a GRH. <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li></ul>Geodesics and coding <ul style="display: flex;"><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Hurwitz continued fractions Definition (Hurwitz continued fractions (CF)) We identify a sequence of integers, a0 ∈ Z, and a1, a2, . . . ∈ Z⋆ with a point −1 <ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">a</li></ul>2x = Ta0 ST 1 ST · · · 0 = a0 + =: [a0; a1, a2, . . .] −1 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”: <ul style="display: flex;"><li style="flex:1">no 1 appear </li><li style="flex:1">and </li><li style="flex:1">if ai = 2 then ai+1 ≶ 0. </li></ul>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 ]. <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li></ul>Geodesics and coding <ul style="display: flex;"><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>Associated dynamical system The generating map f 12<ul style="display: flex;"><li style="flex:1">(x) ∈ Z denotes the nearest integer of x, i.e. |x − (x)| ≤ </li><li style="flex:1">.</li></ul><ul style="display: flex;"><li style="flex:1">ꢆ</li><li style="flex:1">ꢇ</li></ul>1For I = −2 , 12 the generating map for the CF of x is <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">−1 </li><li style="flex:1">−1 </li></ul>f : I → I; x → −.<ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul>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 <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>−1 Ω → Ω; (x, y) → f (x), y + a1 with x = [0; a1, . . .]. <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li></ul>Geodesics and coding <ul style="display: flex;"><li style="flex:1">Transfer operator </li><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>PSL (Z) and geodesics 2Recall PSL2(Z) = hS, Ti. We denote geodesics γ on H by its base points: γ = (γ , γ+). −In the diagram we illustrate the closed geodesic γ = ([0; −3, −4], [0; −4, −3]−1). Theorem Each geodesic γ′ is PSL2(Z)-equivalent with a geodesic γ = (γ , γ+) satisfying (γ , γ−1) ∈ Ω. <ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul>+If γ′ is closed then γ = [0; a1, . . . , an] is regular and −γ+−1 = [0; an, . . . , a1] If γ and υ satisfy (γ , γ−1), (υ , υ−1) ∈ Ω and no base point is <ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><ul style="display: flex;"><li style="flex:1">+</li><li style="flex:1">+</li></ul>√ 3− 5<ul style="display: flex;"><li style="flex:1">PSL2(Z)-equivalent with [0; 3] = </li><li style="flex:1">then γ = υ. </li></ul>2Nearest λ-multiple continued fractions <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li></ul>Transfer operator <ul style="display: flex;"><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>The Ising model and transfer matrices Ernst Ising (1900 – 1998) discussed an 1 − d lattice spin model. Ising-Model Config. space S = { 1}N; left-shift τ : S → S, (τξ)i := ξi+1. <ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">P</li></ul><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul>Total energy: E = −J i=1 ξi ξi+1 + B i=1 ξi with spin interaction J and magnetic field interaction B.) Ernst Ising ≈ 1925. Partition function Zm(A, s) PXm−1 k=0 kA(τ ξ) e−s <ul style="display: flex;"><li style="flex:1">with s = </li><li style="flex:1">and A(ξ) = J ξ0ξ1 + Bξ0. </li></ul>1Temp. Zm(A, s) = ξ∈S; m−periodic free energy = limm→∞ m−1 log Zm(A, s). Ising rewrote Zm as Xe(ξ1, ξ2) · · · e(ξm, ξ1) with e(ξi , ξj ) = e−s(Jξ ξ −Bξ ) .<ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">j</li><li style="flex:1">i</li></ul>Zm(A, s) = ξ1,...,ξ ∈{ 1} mTransfer matrix Ls <ul style="display: flex;"><li style="flex:1">„</li><li style="flex:1">«</li></ul><ul style="display: flex;"><li style="flex:1">`</li><li style="flex:1">´</li></ul>e(+1, +1) e(+1, −1) e(−1, +1) e(−1, −1) <ul style="display: flex;"><li style="flex:1">Ls := </li><li style="flex:1">satisfies Zm(A, s) = trace Lsm </li><li style="flex:1">.</li></ul><ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li></ul>Transfer operator <ul style="display: flex;"><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>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 L acting on functions h: Λ → C is defined by X<ul style="display: flex;"><li style="flex:1">ꢄ</li><li style="flex:1">ꢅ</li></ul><ul style="display: flex;"><li style="flex:1">Lh (x) = </li><li style="flex:1">g(y) h(y) </li></ul>y∈f −1(x) Remarks: Usually, take g = |J|−1 if the Jacobian J of f exists. ꢄꢅ−1 PL of the form Lh(x) = f ′(y) h(y) is also known as a y∈f −1(x) Perron-Frobenius Operator. more Relation of L to the dynamical zeta-function: 1ζ(z) = .detc (1 − zL) <ul style="display: flex;"><li style="flex:1">Front </li><li style="flex:1">Modular surface </li><li style="flex:1">Spectral theory </li><li style="flex:1">Geodesics and coding </li></ul>Transfer operator <ul style="display: flex;"><li style="flex:1">Connection to spectral theory </li><li style="flex:1">Conclusions </li></ul>The transfer operator for continued fractions Recall respectively define: 0 −1 1 0 <ul style="display: flex;"><li style="flex:1">`</li><li style="flex:1">´</li><li style="flex:1">`</li><li style="flex:1">´</li></ul>1 1 0 1 S = <ul style="display: flex;"><li style="flex:1">and T = </li><li style="flex:1">∈ PSL2(Z), </li></ul><ul style="display: flex;"><li style="flex:1">˛</li><li style="flex:1">`</li><li style="flex:1">`</li><li style="flex:1">´´ </li><li style="flex:1">`</li><li style="flex:1">´</li><li style="flex:1">`</li><li style="flex:1">´</li></ul>−s a b c d (z) := (cz + d)2 hand az+b cz+d ˛h2s <ul style="display: flex;"><li style="flex:1">ˆ</li><li style="flex:1">˜</li><li style="flex:1">ˆ</li><li style="flex:1">˜</li></ul>f : −12 , → −12 , ;x → (mod 1) = T−nS x for some n ∈ Z. 12 12 −1 xThe associated transfer operator is formally given by ꢄꢅ−s PLs h(x) := f ′(y) <ul style="display: flex;"><li style="flex:1">h(y) </li><li style="flex:1">(x ∈ [−1/2, 1/2]) </li></ul>y∈f −1(x) Banach space (with sup-norm) V := C(D) ∩ Cω(D◦), D = {z; |z| ≤ 1}. Transfer operator for continued fractions The operator Ls : : V × V → V × V , Re(s) > 1, is defined as <ul style="display: flex;"><li style="flex:1">ꢈ</li><li style="flex:1">ꢈ</li></ul>ꢀP∞ P∞ ꢁn=3 h1ꢈ2sSTn ++n=2 h2ꢈ2s ST−n <ul style="display: flex;"><li style="flex:1">ꢈ</li><li style="flex:1">ꢈ</li></ul>~<ul style="display: flex;"><li style="flex:1">P∞ </li><li style="flex:1">P∞ </li></ul>Ls h(z) =

The Transfer Operator Approach to ``Quantum Chaos

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