Quantum maps from transfer operators
type: Item
instance of: scholarly article
title: Quantum maps from transfer operators
zbMATH Open document ID: 0795.58038
DOI: 10.1016/0167-2789(93)90199-B
author: M. Carioli
author: E. Bogomolny
published in: Physica D
publication date: 1994-09-11T00:00:00Z
review text: The Selberg zeta function \(\zeta_ s(s)\) yields an exact relationship between the periodic orbits of a fully chaotic Hamiltonian system (the geodesic flow on surfaces of constant negative curvature) and the corresponding quantum system (the spectrum of the Laplace-Beltrami operator on the same manifold). It was found that for certain manifolds \(\zeta_ s (s)\) can be exactly rewritten as the Fredholm-Grothendieck determinant \(\text{det (\textbf{1}}-{\mathbf T}_ s)\), where \({\mathbf T}_ s\) is a generalization of the Ruelle-Perron-Frobenius transfer operator. We present an alternative derivation of this result, yielding a method to find not only the spectrum but also the eigenfunctions of the Laplace- Beltrami operator in terms of eigenfunctions of \({\mathbf T}_ s\). Various properties of the transfer operator are investigated both analytically and numerically for several systems.
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 493551
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: spectrum
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Laplace-Beltrami operator
reviewed by: R. Schmid
describes a project that uses: Mathematica
MaRDI profile type: Publication
cites work: Semiclassical quantization of multidimensional systems
cites work: Q3237356
cites work: The Selberg trace formula for \(\text{PSL}(2,\mathbb R)\). Vol. 2
cites work: Chaos in classical and quantum mechanics
cites work: The thermodynamic formalism approach to Selberg’s zeta function for 𝑃𝑆𝐿(2,𝐙)
cites work: La théorie de Fredholm
cites work: Some applications of thermodynamic formalism to manifolds with constant negative curvature
cites work: Q4185090
cites work: Topics in harmonic analysis on homogeneous spaces
cites work: On the thermodynamic formalism for the Gauss map
cites work: Q4002921
cites work: The Modular Surface and Continued Fractions
cites work: The backward continued fraction map and geodesic flow
cites work: Q3331506
cites work: Geodesic flows, interval maps, and symbolic dynamics
cites work: Axiom A Diffeomorphisms have Rational Zeta Functions
cites work: Symbolic Dynamics for Hyperbolic Flows
cites work: Q4744336
description: scientific article; zbMATH DE number 493551