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Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus. (English) Zbl 1202.81076

The author investigates distributions for expectation values of observables for quantized symplectic maps on the \(2d\)-dimensional torus \(\mathbb T^{2d}\) equipped with the standard symplectic structure (generalizations to higher dimensions of the familiar ‘cat map’ on \(\mathbb T^2\)). The classical map is given as an element \(A\) of a subgroup consisting of (defined in the paper) ‘quantizable elements’ of the symplectic group \(\text{Sp}(2d,\mathbb Z)\). The quantization procedure consists of 1) identifying the finite dimensional space \({\mathcal H}_N=L^2(\mathbb Z/N\mathbb Z)^d\) as the Hilbert space of quantum states, 2) quantization of ‘classical observables’, i.e., smooth functions on \(\mathbb T^2\) by mapping them on linear operators acting on \(\text{Op}_N(f)\) acting on \({\mathcal H}_N\) (‘quantum observables’) , 3) ‘quantization of dynamics’, i.e., mapping the classical map \(A\) on a unitary operator \(U_N(A)\) acting on \({\mathcal H}\) (the ‘quantum propagator’). The integer parameter \(N\) plays the role of the inverse Planck constant, hence the classical limit corresponds to \(N\to\infty\). Two additional conditions connecting the classical and quantum systems are imposed on the quantization: 1) the commutator of two quantum observables corresponding to classical observables should tend for \(N\to\infty\), after multiplication by \(-iN\), to the quantum observable corresponding to the Poisson bracket of \(f\) and \(g\) (in the physical literature it is usually called the Dirac quantization condition), and 2) the quantum observable corresponding to a classical observable \(A\circ f\), i.e., \(f\) evolved by the classical symplectic map \(A\), should tend for \(N\to\infty\) to \(U_N^{-1}\text{Op}_N(f)U_N\), i.e., the corresponding quantum observable evolved according to laws of quantum mechanics.
In addition to the quantization, a family of commuting operators commuting with \(U_N(A)\) (the quantum ‘Hecke operators’) is constructed for each \(N\). The common eigenfunctions of all Hecke operators are called Hecke eigenfunctions. Several theorems concerning behavior of quantum expectation values \(\langle\text{Op}_N(f)\psi,\psi\rangle\) for \(N\to\infty\) are proved. The most important states that for a matrix \(A\) with distinct eigenvalues the expectation values for any sequence of \(\psi^{(N)}\) of Hecke eigenfunctions tend to the average of \(f\) with respect to the Lebesgue measure on the torus if and only if \(A\) has no rational invariant subspaces. Maps for which such quantum expectation values converge are called ‘arithmetically quantum uniquely ergodic’. For such maps a bound for convergence rate with \(N\to\infty\), as well as the variance of expectation values calculated for all elements of the Hecke eigenbasis are given. On the contrary, for matrices \(A\) with distinct eigenvalues having invariant rational subspaces there are sequences of Hecke eigenfunctions for which the expectation values converge to distributions localized on corresponding invariant submanifolds of the torus (the, so called, ‘super scars’).

MSC:

81Q50 Quantum chaos
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
11L07 Estimates on exponential sums
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
11F60 Hecke-Petersson operators, differential operators (several variables)
47A35 Ergodic theory of linear operators
81S10 Geometry and quantization, symplectic methods
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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