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An Isaac Newton Institute Workshop

Random Matrix Theory and Arithmetic Aspects of Quantum Chaos

On the distribution of matrix elements for the quantum cat map

Authors: Par Kurlberg (Chalmers), Zeev Rudnick (Tel Aviv University)

Abstract

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. We will study the fluctuations of the matrix elements for the desymmetrized quantum cat map and present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We will show that the second and fourth moment of the distribution agree with the conjecture, and also present some numerical evidence.

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