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48
Ergodic billiards that are not quantum unique ergodic
, 2008
"... Partially rectangular domains are compact twodimensional Riemannian manifolds X, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai bil ..."
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Cited by 30 (0 self)
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Partially rectangular domains are compact twodimensional Riemannian manifolds X, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a oneparameter family Xt of such domains parametrized by the aspect ratio t of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on Xt with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all t ∈ [1,2] excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be nonQUE.
RECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS
, 2009
"... This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigen ..."
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This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the nonQUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss ’ QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question ‘why study matrix elements’ it presents an application of the author to the geometry of nodal sets.
Quantum unique ergodicity for SL2(Z)H
"... An interesting problem in number theory and quantum chaos is to understand the distribution of Maass cusp forms of large Laplace eigenvalue for the modular surface X = SL2(Z)\H. Let φ denote a Maass form of eigenvalue λ, normalized so that its Petersson norm ∫ ..."
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Cited by 20 (1 self)
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An interesting problem in number theory and quantum chaos is to understand the distribution of Maass cusp forms of large Laplace eigenvalue for the modular surface X = SL2(Z)\H. Let φ denote a Maass form of eigenvalue λ, normalized so that its Petersson norm ∫
Real zeros of holomorphic Hecke cusp forms
 J. Eur. Math. Soc
, 2012
"... Abstract. This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of rea ..."
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Cited by 13 (1 self)
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Abstract. This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Secondary: 34F05.
Sieving for mass equidistribution
"... We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Ho], with various simplifications in our analysis due to the knowledge of the RamanujanPetersson conjecture in this holomorphic ca ..."
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We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Ho], with various simplifications in our analysis due to the knowledge of the RamanujanPetersson conjecture in this holomorphic case. 1 Introduction and statement of results We study the shifted convolution sums λ1(n)λ2(n + ℓ) n�x
Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels
 J. Amer. Math. Soc
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