[Paper Review] Value distribution for eigenfunctions of desymmetrized quantum maps

Pär Kurlberg, Zeév Rudnick|ArXiv.org|Jan 8, 2001

Quantum chaos and dynamical systems6 references4 citations

TL;DR

This paper investigates the value distribution and suprema of Hecke eigenfunctions for desymmetrized quantum cat maps, using number-theoretic tools like Weil's bound and Katz's exponential sum theory. For split primes, it proves that amplitudes of nontrivial eigenfunctions converge to a semi-circle distribution and become statistically independent, with optimal $ L^∞ $ bounds of $ \leq 2\sqrt{1 - 1/N} $, while trivial eigenfunctions have constant amplitude.

ABSTRACT

We study the value distribution and extreme values of eigenfunctions for the ``quantized cat map''. This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called ``Hecke operators''. The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call ``Hecke eigenfunctions''. In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the ``split primes'' for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.

Motivation & Objective

• To analyze the extreme values and value distribution of eigenfunctions in quantum chaotic systems, specifically the quantized cat map.
• To understand the behavior of Hecke eigenfunctions—eigenfunctions of both the quantum map and associated Hecke operators—under desymmetrization.
• To establish sharp bounds on the $ L^\infty $-norm of these eigenfunctions for general $ N $, and exact distributional limits for split primes.
• To explore statistical independence of amplitudes across different eigenfunctions in the large-$ N $ limit.

Proposed method

• Use of Hecke operators to construct orthonormal bases of eigenfunctions that diagonalize the quantum map, enabling the study of eigenfunction properties.
• Application of Weil’s bound on exponential sums to control the size of Hecke eigenfunctions, especially for prime $ N $.
• Leveraging N. Katz’s theorem on the value distribution of normalized exponential sums $ F_p(\chi, R)(t) $, which are related to the eigenfunctions via a scaling transformation.
• Derivation of the $ L^\infty $-norm bound via Gauss sum estimates and the structure of the Bruhat decomposition of the diagonalizing matrix.
• Relating the absolute values of eigenfunctions to normalized exponential sums $ F_p(\chi, R)(t) $, which are known to equidistribute in $[-1,1]$ with respect to the semi-circle measure.
• Using a normalization lemma to transfer the value distribution of $ F_p $ to the eigenfunctions $ \psi_{\chi,N} $, showing that $ |\psi_{\chi,N}|/2 $ converges to the semi-circle law.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the $ L^\infty $-norm of Hecke eigenfunctions for the desymmetrized quantum cat map?
• RQ2 How do the amplitudes of Hecke eigenfunctions distribute over the torus as $ N \to \infty $, particularly for split primes?
• RQ3 Are different Hecke eigenfunctions statistically independent in the large-$ N $ limit?
• RQ4 Can the value distribution of eigenfunctions be described using classical results from exponential sums over finite fields?

Key findings

• For general $ N $, the $ L^\infty $-norm of any Hecke eigenfunction satisfies $ \|\psi\|_{\infty} \ll_\epsilon N^{3/8 + \epsilon} $ for all $ \epsilon > 0 $, improving on the trivial $ N^{1/2} $ bound.
• For split primes $ N = p $, the eigenfunctions corresponding to the trivial character have constant amplitude: $ |\psi_{0,p}| = \sqrt{1 - 1/p} $, and $ |\psi_{\infty,p}| = 1 $.
• For nontrivial characters $ \chi $, the $ L^\infty $-norm is uniformly bounded: $ \|\psi_{\chi,p}\|_{\infty} \leq 2\sqrt{1 - 1/p} $.
• The normalized amplitudes $ |\psi_{\chi,p}(t)|/2 $ converge in distribution to the semi-circle measure $ \mu_{sc}(I) = \int_I \frac{4}{\pi}\sqrt{1 - u^2} du $ as $ p \to \infty $.
• For $ r \geq 2 $, the amplitudes $ |\psi_{\chi_1,p}|, \dots, |\psi_{\chi_r,p}| $ become statistically independent in the $ p \to \infty $ limit, with joint distribution converging to the product of semi-circle measures.
• The value distribution result is derived via a transformation of the eigenfunction into a normalized exponential sum $ F_p(\chi, R)(t) $, whose distribution is governed by Katz’s theorem on equidistribution with respect to the semi-circle law.