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  • Modular surface
  • Spectral theory
  • Geodesics and coding
  • Transfer operator
  • Connection to spectral theory
  • Conclusions

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The transfer operator approach to “quantum chaos”

Classical mechanics and the Laplace-Beltrami operator on PSL2(Z)\H

Tobias Mu¨hlenbruch
Joint work with D. Mayer and F. Str¨omberg

Institut fu¨r 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

23456

Spectral theory Geodesics and coding Transfer operator Connection to spectral theory Conclusions

Front

Modular surface

  • Spectral theory
  • Geodesics and coding
  • Transfer operator
  • Connection to spectral theory
  • Conclusions

The full modular group PSL (Z)
2

Let PSL2(Z) be the full modular group

PSL2(Z) = hS, T|(ST)3 = 1i = SL2(Z) mod { 1}

  • ꢀ
  • ꢁ
  • ꢀ
  • ꢁ
  • ꢀ
  • ꢁ

01

  • −1
  • 1

0
11
10
01

  • with S =
  • , T =
  • and 1 =
  • .

0

  • ꢀ
  • ꢁ
  • ꢂ

az+b cz+d

  • a
  • b

if z ∈ C,

  • M¨obius transformations:
  • z =

a

  • c
  • d

if z = ∞.

c

Three 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

  • translation T : z → z + 1 and inversion S : z →
  • .

z

ꢃ

12

(Closed) fundamental domain F = z ∈ H; |z| ≥ 1, |Re(z)| ≤

.

Front

Modular surface

  • Spectral theory
  • Geodesics and coding
  • Transfer operator
  • Connection to spectral theory
  • Conclusions

The modular surface PSL (Z)\H
2

The 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 M¨obius 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
3

cusp at z → i∞ and two conic singularities at z = i and z = The hyperbolic plane H is also called the Poincar´e half plane. PSL2(Z)\H is arithmetic (  connection to number theory).
.

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

Gq = hS, Tq|(STq)q = 1i with
, λq = 2 cos π/q and q = 3, 4, 5, . . ..

  • ꢀ
  • ꢁ

  • ꢄ
  • ꢅ

10

λq

1

T =

Example: PSL2(Z) = G3. M¨obius transformation extends to Gq. Fundamental domain

  • n
  • o

λq

Fq = z ∈ H; |z| ≥ 1, |Re(z)| ≤

.

2

Fundamental domain forms a (2, q, ∞)

π

2

π

q

  • hyperbolic triangle with angles
  • ,
  • and 0.

Gq\H is finite and non-compact. Gq\H is non-arithmetic for q = 3, 4, 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,

  • ꢄ
  • ꢅ

  • ∆u = s(1 − s) u with ∆ = −y2 ∂x2 + ∂y2
  • ,

u(M z) = u(z) for all M ∈ PSL2(Z),

  • ꢄ
  • ꢅ

u(x + iy) = O yC as y → ∞ for all C ∈ R.

Maass cusp form at

1

s = + i13.7797 . . ..

∆ admits self-adjoined extension in L2 PSL2(Z)\H .

2

  • ꢄ
  • ꢅ

s is called spectral parameter.

s(1 − s) > 14 , i.e., s ∈ + iR⋆.

12

Discrete 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) =

  • y
  • an K

1

s− 2

0=n∈Z

12

s = + i9.533 . . ..

  • Front
  • Modular surface

Spectral theory

  • Geodesics and coding
  • Transfer operator
  • Connection to spectral theory
  • Conclusions

The Laplace-Beltrami operator

  • ꢄ
  • ꢅ

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

R

2

F |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.

  • 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
−1

  • a
  • a

2

x = Ta0 ST 1 ST · · · 0 = a0 +

=: [a0; a1, a2, . . .]

−1

a1 +

−1

...

a

2

+

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
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  • Spectral theory

Geodesics and coding

  • Transfer operator
  • Connection to spectral theory
  • Conclusions

Associated dynamical system

The generating map f

12

  • (x) ∈ Z denotes the nearest integer of x, i.e. |x − (x)| ≤
  • .

  • ꢆ
  • ꢇ

1

For I = −2 , 12 the generating map for the CF of x is

  • ꢀ
  • ꢁ

  • −1
  • −1

f : I → I; x →

−

.

  • 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),

y + a1

with x = [0; a1, . . .].

  • Front
  • Modular surface
  • Spectral theory

Geodesics and coding

  • Transfer operator
  • Connection to spectral theory
  • Conclusions

PSL (Z) and geodesics
2

Recall 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.

  • −
  • −

+

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

  • −
  • −

  • +
  • +

√
3−

5

  • PSL2(Z)-equivalent with [0; 3] =
  • then γ = υ.

2

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

Config. space S = { 1}N; left-shift τ : S → S, (τξ)i := ξi+1.

  • P
  • P

  • ∞
  • ∞

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)

P

X

m−1 k=0

k

A(τ ξ)

e−s

  • with s =
  • and A(ξ) = J ξ0ξ1 + Bξ0.

1Temp.

Zm(A, s) =

ξ∈S; m−periodic

  free energy = limm→∞ m−1 log Zm(A, s).
Ising rewrote Zm as

X

e(ξ1, ξ2) · · · e(ξm, ξ1) with e(ξi , ξj ) = e−s(Jξ ξ −Bξ )

.

  • i
  • j
  • i

Zm(A, s) =

ξ

1

,...,ξ ∈{ 1}

m

Transfer matrix Ls

  • „
  • «

  • `
  • ´

e(+1, +1) e(+1, −1) e(−1, +1) e(−1, −1)

  • Ls :=
  • satisfies Zm(A, s) = trace Lsm
  • .

  • 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 L acting on functions h: Λ → C is defined by

X

  • ꢄ
  • ꢅ

  • Lh (x) =
  • g(y) h(y)

y∈f −1(x)

Remarks:
Usually, take g = |J|−1 if the Jacobian J of f exists.

ꢄ

ꢅ−1

P

L 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)

  • 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 0

  • `
  • ´
  • `
  • ´

1 1 0 1

S =

  • and T =
  • ∈ PSL2(Z),

  • ˛
  • `
  • `
  • ´´
  • `
  • ´
  • `
  • ´

−s a b

c d

(z) := (cz + d)2

h

and

az+b cz+d

˛

h

2s

  • ˆ
  • ˜
  • ˆ
  • ˜

f : −12 ,

→ −12 ,

;

x →

(mod 1) = T−nS x for some n ∈ Z.

12
12
−1

x

The associated transfer operator is formally given by

ꢄ

ꢅ−s

P

Ls h(x) :=

f ′(y)

  • h(y)
  • (x ∈ [−1/2, 1/2])

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

  • ꢈ
  • ꢈ

ꢀP∞

P∞

ꢁ

n=3 h1ꢈ2sSTn

++

n=2 h2ꢈ2s ST−n

  • ꢈ
  • ꢈ

~

  • P∞
  • P∞

Ls h(z) =

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  • Introduction to the Transfer Operator Method

    Introduction to the Transfer Operator Method

    Omri Sarig Introduction to the transfer operator method Winter School on Dynamics Hausdorff Research Institute for Mathematics, Bonn, January 2020 Contents 1 Lecture 1: The transfer operator (60 min) ......................... 3 1.1 Motivation . .3 1.2 Definition, basic properties, and examples . .3 1.3 The transfer operator method . .5 2 Lecture 2: Spectral Gap (60 min) ................................ 7 2.1 Quasi–compactness and spectral gap . .7 2.2 Sufficient conditions for quasi-compactness . .9 2.3 Application to continued fractions. .9 3 Lecture 3: Analytic perturbation theory (60 min) . 13 3.1 Calculus in Banach spaces . 13 3.2 Resolvents and eigenprojections . 14 3.3 Analytic perturbations of operators with spectral gap . 15 4 Lecture 4: Application to the Central Limit Theorem (60 min) . 17 4.1 Spectral gap and the central limit theorem . 17 4.2 Background from probability theory . 17 4.3 The proof of the central limit theorem (Nagaev’s method) . 18 5 Lecture 5 (time permitting): Absence of spectral gap (60 min) . 23 5.1 Absence of spectral gap . 23 5.2 Inducing . 23 5.3 Operator renewal theory . 24 A Supplementary material ........................................ 27 A.1 Conditional expectations and Jensen’s inequality . 27 A.2 Mixing and exactness for the Gauss map . 30 A.3 Hennion’s theorem on quasi-compactness . 32 A.4 The analyticity theorem . 40 A.5 Eigenprojections, “separation of spectrum”, and Kato’s Lemma . 41 A.6 The Berry–Esseen “Smoothing Inequality” . 43 1 Lecture 1 The transfer operator 1.1 Motivation A thought experiment Drop a little bit of ink into a glass of water, and then stir it with a tea spoon.
  • Reproducing Kernel Hilbert Space Compactification of Unitary Evolution Groups

    Reproducing Kernel Hilbert Space Compactification of Unitary Evolution Groups

    Appl. Comput. Harmon. Anal. 54 (2021) 75–136 Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha Reproducing kernel Hilbert space compactification of unitary evolution groups Suddhasattwa Das a, Dimitrios Giannakis a,∗, Joanna Slawinska b a Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA b Finnish Center for Artificial Intelligence, Department of Computer Science, University of Helsinki, Helsinki, FI-00014, Finland a r t i c l e i n f o a b s t r a c t Article history: A framework for coherent pattern extraction and prediction of observables of Received 1 April 2019 measure-preserving, ergodic dynamical systems with both atomic and continuous Received in revised form 23 spectral components is developed. This framework is based on an approximation November 2020 of the generator of the system by a compact operator W on a reproducing Accepted 26 February 2021 τ kernel Hilbert space (RKHS). The operator W is skew-adjoint, and thus can Available online 3 March 2021 τ Communicated by E. Le Pennec be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in Keywords: terms of a Dirichlet energy, and provide a notion of coherent observables under the Koopman operators dynamics akin to the Koopman eigenfunctions associated with the atomic part of tWτ Perron-Frobenius operators the spectrum. In addition, Wτ generates a unitary evolution group {e }t∈R on the Ergodic dynamical systems RKHS, which approximates the unitary Koopman group of the system.
  • DYNAMICAL SYSTEMS, TRANSFER OPERATORS and FUNCTIONAL

    DYNAMICAL SYSTEMS, TRANSFER OPERATORS and FUNCTIONAL

    DYNAMICAL SYSTEMS, TRANSFER OPERATORS and FUNCTIONAL ANALYSIS Brigitte Vallee´ , Laboratoire GREYC (CNRS et Universit´ede Caen) S´eminaire CALIN, LIPN, 5 octobre 2010 A Euclidean Algorithm ⇓ Arithmetic properties of the division ⇓ Geometric properties of the branches ⇓ Spectral properties of the transfer operator ⇓ Analytical properties of the Quasi-Inverse of the transfer operator ⇓ Analytical properties of the generating function ⇓ Probabilistic analysis of the Euclidean Algorithm Dynamical analysis of a Euclidean Algorithm. Geometric properties of the branches ⇓ Spectral properties of the transfer operator ⇓ Analytical properties of the Quasi-Inverse of the transfer operator ⇓ Analytical properties of the generating function ⇓ Probabilistic analysis of the Euclidean Algorithm Dynamical analysis of a Euclidean Algorithm. A Euclidean Algorithm ⇓ Arithmetic properties of the division ⇓ ⇓ Analytical properties of the generating function ⇓ Probabilistic analysis of the Euclidean Algorithm Dynamical analysis of a Euclidean Algorithm. A Euclidean Algorithm ⇓ Arithmetic properties of the division ⇓ Geometric properties of the branches ⇓ Spectral properties of the transfer operator ⇓ Analytical properties of the Quasi-Inverse of the transfer operator Dynamical analysis of a Euclidean Algorithm. A Euclidean Algorithm ⇓ Arithmetic properties of the division ⇓ Geometric properties of the branches ⇓ Spectral properties of the transfer operator ⇓ Analytical properties of the Quasi-Inverse of the transfer operator ⇓ Analytical properties of the generating function ⇓ Probabilistic analysis of the Euclidean Algorithm u0 := v; u1 := u; u0 ≥ u1 u0 = m1u1 + u2 0 < u2 < u1 u = m u + u 0 < u < u 1 2 2 3 3 2 ... = ... + up−2 = mp−1up−1 + up 0 < up < up−1 up−1 = mpup + 0 up+1 = 0 up is the gcd of u and v, the mi’s are the digits.
  • Conservative L-Systems and the Livšic Function 1

    Conservative L-Systems and the Livšic Function 1

    CONSERVATIVE L-SYSTEMS AND THE LIVSICˇ FUNCTION S. BELYI, K. A. MAKAROV, AND E. TSEKANOVSKI˘I Dedicated to Yury Berezansky, a remarkable Mathematician and Human Being, on the occasion of his 90th birthday. Abstract. We study the connection between the classes of (i) Livˇsic functions s(z), i.e., the characteristic functions of densely defined symmetric operators A˙ with de- ficiency indices (1, 1); (ii) the characteristic functions S(z) of a maximal dissipative extension T of A,˙ i.e., the M¨obius transform of s(z) determined by the von Neu- mann parameter κ of the extension relative to an appropriate basis in the deficiency subspaces; and (iii) the transfer functions WΘ(z) of a conservative L-system Θ with the main operator T . It is shown that under a natural hypothesis the functions S(z) 1 − 1 and WΘ(z) are reciprocal to each other. In particular, WΘ(z)= S(z) = s(z) when- ever κ = 0. It is established that the impedance function of a conservative L-system with the main operator T belongs to the Donoghue class if and only if the von Neu- mann parameter vanishes (κ = 0). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. We also obtain the representation of a function from this class via the Weyl-Titchmarsh function. All results are illustrated by a number of examples. 1. Introduction Suppose that T is a densely defined closed operator in a Hilbert space such that H its resolvent set ρ(T ) is not empty and assume, in addition, that Dom(T ) Dom(T ∗) is ∩ ˙ ∗ dense.