Abstract
Abstract
We study ℓ
∞ norms of ℓ
2-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000, Communications in Mathematical Physics, 211, 659–686)) we show that there exists a sequence of eigenfunctions u with
∥
u
∥
∞
≳
log
N
−
1
/
2
. For general eigenfunctions we show the upper bound
∥
u
∥
∞
≲
log
N
−
1
/
2
. Here the semiclassical parameter is
h
=
2
π
N
−
1
. Our upper bound is analogous to the one proved by Bérard in P Bérard (1977, Mathematische Zeitschrift, 155, 249-276) for compact Riemannian manifolds without conjugate points.
Funder
Division of Mathematical Sciences
Division of Graduate Education
Massachusetts Institute of Technology
Subject
Condensed Matter Physics,Mathematical Physics,Atomic and Molecular Physics, and Optics
Reference18 articles.
1. Quantization of linear maps-fresnel diffraction by a periodic grating;Berry;Physica D,1980
2. The behaviour of eigenstates of arithmetic hyperbolic manifolds;Rudnick;Commun. Math. Phys.,1994
3. On the asymptotic behavior of a spectral function and on expansion in eigenfunctions of a self-adjoint differential equation of second order;Levitan;Izv. Akad. Nauk SSSR Ser. Mat.,1955
4. Über die eigenfunktionen auf geschlossenen riemannschen mannigfaltigkeiten;Avakumovic;Math. Z.,1956
5. The spectral function of an elliptic operator;Hörmander;Acta Math.,1968