Heisenberg Realizations, Eigenfunctions and Proof of the Kurlberg-Rudnick Supremum Conjecture

2005/11/10 by Shamgar Gurevich, Ronny Hadani, Gurevich, Shamgar +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT #quant-ph

paper · pdf · doi:10.48550/arxiv.math-ph/0511036

Preliminary version, 31 pages

openalex publication_date 2005/11/10 · arxiv created 2005/11/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, proof of the \it Kurlberg-Rudnick supremum conjecture for the quantum Hannay-Berry model is presented. This conjecture was stated in P. Kurlberg's lectures at Bologna 2001 and Tel-Aviv 2003. The proof is a primer application of a fundamental solution: all the Hecke eigenfunctions of the quantum system are constructed. The main tool in our construction is the categorification of the compatible system of realizations of the Heisenberg representation over a finite field. This enables us to construct certain "perverse sheaves" that stands motivically prior to the Hecke eigenfunctions.

Cited by

• Notes on quantization of symplectic vector spaces over finite fields
• Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation

Related

• The Higher-Dimensional Rudnick-Kurlberg Conjecture
• Proof of the Kurlberg-Rudnick Rate Conjecture
• The Two Dimensional Hannay-Berry Model
• Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group
• Weyl's Relations, Integrable Matrix Models and Quantum Computation