<i>Quantum Chaos: An Introduction</i>

Stöckmann, H.‐J.

doi:10.1119/1.19544

Cited by 417 publications

(813 citation statements)

References 0 publications

Supporting

Mentioning

795

Contrasting

Unclassified

Order By: Relevance

“…Among the many measures that have been applied to characterize these statistical features, much attention has been given to two-point correlation functions since they can under certain assumptions be related to the classical dynamics [4]. For the case of spectra of bounded systems this has worked remarkably well and in addition one of the main predictions of the semiclassical analysis, the existence of long range correlations due to periodic orbits [4,5], has been confirmed many times [1,6].…”

Section: Introductionmentioning

confidence: 99%

Semiclassical cross section correlations

2000

View full text Add to dashboard Cite

We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth classical limit the autocorrelation function of such matrix elements has two contributions with relative weights determined by classical dynamics. We show how the random matrix result can be obtained if the operator approaches a projector onto a single initial state. The expressions are verified in calculations for the kicked rotor.

show abstract

Section: Introductionmentioning

confidence: 99%

Semiclassical cross section correlations

2000

View full text Add to dashboard Cite

show abstract

“…On a bounded region Q ⊂ R 2 (the billiard table), an infinitesimal particle moves along segments at unit speed, changing direction according to the law of specular reflection upon collisions at boundaries. The essential link in billiards between the geometry of the table and the dynamics of the system facilitates a robust model which has proved useful in approaching problems ranging from the foundations of the Boltzmann's ergodic hypothesis [9], to the description of shell effects in semiclassical physics [5], to the design of microwave resonators in quantum chaos [33], and many other other applications [1,17,23,25,26]. In particular, ergodic properties are determined by the shape of the table, producing a spectrum of behaviors from completely integrable to strongly chaotic.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity in Umbrella Billiards

Correia¹,

Cox²,

Zhang³

2017

NHMP

View full text Add to dashboard Cite

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is 0. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to non-ergodic lemon and moon billiards, small increases in the new parameter transform elliptic 2-periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.

show abstract

“…Zaslavsky and G.P.Berman [8] were the first who considered the equation of the Mathieu-Schrodinger for the quantum description of the nonlinear resonance in the approximation of moderate nonlinearity. They studied the case of quasi-classical approximation ∆I ≫ for both variables I 0 and ∆I.…”

Section: Having Written the Stationary Schrodinger Equationŝmentioning

confidence: 99%

“…The key point is that, study of quantum chaos as a rule is focused on the semiclassical domain. We mean not only Gutzwiller's semi-classical path integration method [8], but also RMT. Since the RMT, in somehow implies semi-classical limit.…”

Section: Introductionmentioning

confidence: 99%

Quantum chaos and its kinetic stage of evolution

Chotorlishvili

Ugulava

2010

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

Usually reason of irreversibility in open quantum-mechanical system is interaction with a thermal bath, consisting form infinite number of degrees of freedom. Irreversibility in the system appears due to the averaging over all possible realizations of the environment states. But, in case of open quantum-mechanical system with few degrees of freedom situation is much more complicated.Should one still expect irreversibility, if external perturbation is just an adiabatic force without any random features? Problem is not clear yet. This is main question we address in this review paper. We prove that key point in the formation of irreversibility in chaotic quantum-mechanical systems with few degrees of freedom, is the complicated structure of energy spectrum. We shall consider quantum mechanical-system with parametrically dependent energy spectrum. In particular, we study energy spectrum of the Mathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial, consists from the domains of non-degenerated and degenerated stats, separated from each other by branch points. Due to the modulation of the parameter, system will perform transitions from one domain to other one. For determination of eigenstates for each domain and transition probabilities between them, we utilize methods of abstract algebra. We shall show that peculiarity of parametrical dependence of energy terms, leads to the formation of mixed state and to the irreversibility, even for small number of levels involved into the process. This last statement is important. Meaning is that, we are going to investigate quantum chaos in essentially quantum domain.In the second part of the paper, we will introduce concept of random quantum phase approximation. Then along with the methods of random matrix theory, we will use this assumption, for derivation of muster equation in the formal and mathematically strict way.

show abstract

Quantum Chaos: An Introduction

Cited by 417 publications

References 0 publications

Semiclassical cross section correlations

Semiclassical cross section correlations

Ergodicity in Umbrella Billiards

Quantum chaos and its kinetic stage of evolution

Contact Info

Product

Resources

About