437 research outputs found
On the diagonalization of the discrete Fourier transform
The discrete Fourier transform (DFT) is an important operator which acts on
the Hilbert space of complex valued functions on the ring Z/NZ. In the case
where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors
for the DFT. The transition matrix from the standard basis to the canonical
basis defines a novel transform which we call the discrete oscillator transform
(DOT for short). Finally, we describe a fast algorithm for computing the
discrete oscillator transform in certain cases.Comment: Accepted for publication in the journal "Applied and Computational
Harmonic Analysis": Appl. Comput. Harmon. Anal. (2009),
doi:10.1016/j.acha.2008.11.003. Key words: Discrete Fourier Transform, Weil
Representation, Canonical Eigenvectors, Oscillator Transform, Fast Oscillator
Transfor
Proof of the Kurlberg-Rudnick Rate Conjecture
In this paper we present a proof of the {\it Hecke quantum unique ergodicity
rate conjecture} for the Berry-Hannay model. A model of quantum mechanics on
the 2-dimensional torus. This conjecture was stated in Z. Rudnick's lectures at
MSRI, Berkeley 1999 and ECM, Barcelona 2000.Comment: In this version we add a proof that the character sheaf of the
Heisenberg-Weil representation is perverse, geometrically irreducible of pure
weight 0. Moreover, we supply invariant formulas for the character sheaf on
an appropriate open set, and we give also another alternative proof for the
rate conjecture that uses our invariant formula
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