Classical limit of the quantized hyperbolic toral automorphisms

Abstract

The canonical quantization of any hyperbolic symplectomorphism A of the 2-torus yields a periodic unitary operator on a JV-dimensional Hubert space, N =•£. We prove that this quantum system becomes ergodic and mixing at the classical limit (N -• oo, N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly spread in phase space. Contents 1. Von Neumann definition of the quantum ergodicity and mixing properties. Statement of the main results 473 2. Koopman operator on invariant lattices and periodic orbits. Limits of atomic invariant measures supported on periodic orbits via Kloosterman sums 477 3. Quantization of toral automorphisms. Discrete Wigner functions. Support on classical periodic orbits, relation with the Koopman operator and explicit construction of the quantum eigenvectors 482 4. Classical limit of the matrix elements of the observables via Weil-Deligne exponential sums. Weak-* convergence of the Wigner functions. Proof of the main results 495 Appendix A. Some basic results out of number theory 503

Proof of the Kurlberg-Rudnick Rate Conjecture

“…The first source is computer simulations [Ku] accomplished over the years to give extremely precise bounds for considerably large values of p. A more mathematical argument is based on the fact that for special values of p, in which the Hecke torus splits, namely, T A ≃ F * p , one is able to compute explicitly the eigenstate Ψ ∈ H χ and as a consequence to give an explicit formula for the Wigner distribution [KR2,DGI]. More precisely, in case ξ ∈ T ∨ , i.e., a character, the distribution W χ (ξ) turns out to be equal to the exponential sum:…”

Torus as phase space: Weyl quantization, dequantization, and Wigner formalism

“…We follow the approach in [3], with a suitable modification of the definition, that takes into account the fact that the representation T θ,N is not N -periodic. Then we compute explicitly the Wigner transform and determine its support.…”

“…(i) (Section 2) As in [3] the Weyl quantization of a sufficiently smooth function α on T 2 (classical observable) is defined by replacing exponentials in the Fourier series of α by their representations in C N , and depends on the parameter θ = (θ 1 , θ 2 ) ∈ T 2 labeling the chosen representation. The corresponding Weyl operator A (N × N matrix) is explicitly computed, and depends only on the values of α on the lattice L(θ, N ) := j 2N + θ1 N , k 2N + θ2 N : j, k ∈ Z 2N (Theorem 2.3).…”

“…The construction of the Wigner transform W θ,N ψ of a quantum state ψ ∈ C N on T 2 is presented here starting from any chosen representation of the discrete Heisenberg group following the approach in [3], with some suitable modifications. The resulting W θ,N ψ is shown to be a distribution (signed measure) on phase space with total mass equal to one and support L(θ, N ), (Theorem 3.4); moreover the corresponding marginals are the position and momentum probability distributions with support…”

Quantum Leaks

“…Such exceptional subsequences have been observed in numerical experiments and are referred to as scars or bouncing ball modes. Following earlier results for quantum maps [11,26,20,21], recent seminal contributions on the question of quantum unique ergodicity include the work of Faure, Nonnenmacher and De Bièvre [15,14] who prove the existence of localized eigenstates for quantum cat maps, and Lindenstrauss' proof [25] of quantum unique ergodicity in the case of Hecke eigenstates 3 of the Laplacian on compact arithmetic hyperbolic surfaces of congruence type.…”