Matrix elements for the quantum cat map: fluctuations in short windows

Abstract

We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck's constant. We show that if the length of the window is smaller than the square root of Planck's constant, but larger than the separation between distinct eigenphases, then the variance of this sum is proportional to the length of the window, with a proportionality constant which coincides with the variance of the individual matrix elements corresponding to Hecke eigenfunctions.

The variance of arithmetic measures associated to closed geodesics on the modular surface

“…It points yet again, just as for the local spectral statistics (see the survey [38]), to the source of the singular behaviour of the quantum fluctuations in arithmetic surfaces being the high multiplicity of the length spectrum. Similar phenomena are found for the quantized cat map [18,19].…”

Distributional properties of powers of matrices

“…The case n = 2 seems to be a distinguished one. It was shown in [15] and [14] (see also [13]) to be related with quantum ergodicity on flat tori (an instance of arithmetic quantum chaos). In [23] it is also studied in connection with the order of the reduction of units in quadratic fields.…”

Quantum variance on quaternion algebras, I

“…Bounds for projections. For Φ ∈ A(2 ∞ ) satisfying (35) and A > 0, we claim that Λ(χ, 0) Φ χ ≪ Φ,A C(χ) −A for all χ ∈ X u . To see this, let U be an open subgroup of SL 2 (Q 2 ) that fixes Φ.…”

“…is SL 2 (Q 2 )-invariant. By duality, for Φ ∈ A(2 ∞ ) satisfying (35) there is a unique Φ χ ∈ I(χ) so that Eis(f ), Φ = f, Φ χ (the first inner product taken in A(2 ∞ ), the second in I(χ)). The maps Φ → Φ χ are linear and equivariant for ∆ and SL 2 (Q 2 ).…”