scholarly journals Quantum Limits of Eisenstein Series and Scattering States

2013 ◽  
Vol 56 (4) ◽  
pp. 814-826 ◽  
Author(s):  
Yiannis N. Petridis ◽  
Nicole Raulf ◽  
Morten S. Risager
Keyword(s):  
Eisenstein Series ◽  
Critical Line ◽  
Unique Ergodicity ◽  
Modular Surface ◽  
Scattering States ◽  
Quantum Limits ◽  
Ergodicity Theorem

Abstract.We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Erratum to ‘Quantum Limits of Eisenstein Series and Scattering States’

2013 ◽  
Vol 56 (4) ◽  
pp. 827-828 ◽  
Author(s):  
Yiannis N. Petridis ◽  
Nicole Raulf ◽  
Morten S. Risager
Keyword(s):  
Eisenstein Series ◽  
Scattering States ◽  
Quantum Limits

Exponential Growth Rates of Periodic Asymmetric Oscillators

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Meirong Zhang ◽  
Zhe Zhou
Keyword(s):  
Lyapunov Exponents ◽  
Ergodic Theory ◽  
Growth Rates ◽  
Exponential Growth ◽  
Zero Solution ◽  
Unique Ergodicity ◽  
Topological Dynamics ◽  
Asymmetric Oscillator ◽  
Ergodicity Theorem

AbstractIn this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + qdoes exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

scholarly journals Sign Changes of the Eisenstein Series on the Critical Line

2017 ◽  
Vol 2019 (3) ◽  
pp. 641-672 ◽  
Author(s):  
Junehyuk Jung ◽  
Matthew P Young
Keyword(s):  
Eisenstein Series ◽  
Critical Line ◽  
Sign Changes

scholarly journals EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE

2018 ◽  
Vol 19 (2) ◽  
pp. 581-596 ◽  
Author(s):  
Valentin Blomer
Keyword(s):  
Eisenstein Series ◽  
Quadratic Forms ◽  
Critical Line ◽  
Zeta Functions ◽  
Subconvexity Bounds

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

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