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

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.

2013 ◽  
Vol 56 (4) ◽  
pp. 827-828 ◽  
Author(s):  
Yiannis N. Petridis ◽  
Nicole Raulf ◽  
Morten S. Risager

2008 ◽  
Vol 8 (4) ◽  
Author(s):  
Meirong Zhang ◽  
Zhe Zhou

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.


2017 ◽  
Vol 2019 (3) ◽  
pp. 641-672 ◽  
Author(s):  
Junehyuk Jung ◽  
Matthew P Young

2018 ◽  
Vol 19 (2) ◽  
pp. 581-596 ◽  
Author(s):  
Valentin Blomer

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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