scholarly journals Quantum evolution and classical flow in complex phase space

1990 ◽  
Vol 128 (2) ◽  
pp. 393-409 ◽  
Author(s):  
S. Graffi ◽  
A. Parmeggiani
Keyword(s):  
Phase Space ◽  
Quantum Evolution ◽  
Complex Phase

scholarly journals Tunneling mechanism due to chaos in a complex phase space

2001 ◽  
Vol 64 (2) ◽  
Author(s):  
T. Onishi ◽  
A. Shudo ◽  
K. S. Ikeda ◽  
K. Takahashi
Keyword(s):  
Phase Space ◽  
Complex Phase ◽  
Tunneling Mechanism

scholarly journals Chaotic systems in complex phase space

Pramana ◽  
2009 ◽  
Vol 73 (3) ◽  
pp. 453-470 ◽  
Author(s):  
Carl M. Bender ◽  
Joshua Feinberg ◽  
Daniel W. Hook ◽  
David J. Weir
Keyword(s):  
Phase Space ◽  
Chaotic Systems ◽  
Complex Phase

scholarly journals Local scaling asymptotics in phase space and time in Berezin–Toeplitz quantization

2014 ◽  
Vol 25 (06) ◽  
pp. 1450060 ◽  
Author(s):  
Roberto Paoletti
Keyword(s):  
Phase Space ◽  
Evolution Operator ◽  
Quantum Evolution ◽  
Classical Dynamics ◽  
Semiclassical Asymptotics ◽  
Local Scaling ◽  
Space And Time

This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin–Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time.

Canonical Transformations, Entropy and Quantization

1987 ◽  
Vol 42 (4) ◽  
pp. 333-340 ◽  
Author(s):  
B. Bruhn
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Arbitrary Function ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Integrable Hamiltonian Systems ◽  
Eigenvalue Spectrum ◽  
Complex Phase ◽  
Algebraic Treatment ◽  
Canonical Coordinate

This paper considers various aspects of the canonical coordinate transformations in a complex phase space. The main result is given by two theorems which describe two special families of mappings between integrable Hamiltonian systems. The generating function of these transformations is determined by the entropy and a second arbitrary function which we take to be the energy function. For simple integrable systems an algebraic treatment based on the group properties of the canonical transformations is given to calculate the eigenvalue spectrum of the energy.

scholarly journals A Brief Introduction to Stationary Quantum Chaos in Generic Systems

2020 ◽  
Vol 23 (2) ◽  
pp. 172-191 ◽  
Author(s):  
Marko Robnik
Keyword(s):  
Phase Space ◽  
Time Scale ◽  
Quantum Chaos ◽  
Mixed Type ◽  
Localized States ◽  
Quantum Evolution ◽  
Density Level ◽  
Level Spacing ◽  
Transport Time ◽  
Classical Phase Space

We review the basic aspects of quantum chaos (wave chaos) in mixed-type Hamiltonian systems with divided phase space, where regular regions containing the invariant tori coexist with the chaotic regions. The quantum evolution of classically chaotic bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, as the motion is always almost periodic. However, the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions or Husimi functions) reveals precise analogy of the structure of the classical phase portrait. In classically integrable regions the spectral (energy) statistics is Poissonian, while in the ergodic chaotic regions the random matrix theory applies. If we have the mixed-type classical phase space, in the semiclassical limit (short wavelength approximation) the spectrum is composed of Poissonian level sequence supported by the regular part of the phase space, and chaotic sequences supported by classically chaotic regions, being statistically independent of each other, as described by the Berry-Robnik distribution. In quantum systems with discrete energy spectrum the Heisenberg time tH = πℏ/ΔE, where ΔE is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (transport time) in relation to the Heisenberg time scale tH (their ratio is the parameter α = tH / tT ) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to the normalized inverse participation ratio. We study the structure of quantum localized chaotic eigenstates (their Wigner and Husimi functions) and the distribution of localization measure A. The latter one is well described by the beta distribution, if there are no sticky regions in the classical phase space. Otherwise, they have a complex nonuniversal structure. We show that the localized chaotic states display the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝ S β for small S , where 0 ≤ β ≤ 1, and β = 1 corresponds to completely extended states, and β = 0 to the maximally localized states. β goes from 0 to 1 when α goes from 0 to ∞, β is a function of <A>, as demonstrated in the quantum kicked rotator, the stadium billiard, and a mixed-type billiard.

Lie Series and Canonical Transformations in Complex Phase Space

1988 ◽  
Vol 43 (5) ◽  
pp. 411-418 ◽  
Author(s):  
B. Bruhn
Keyword(s):  
Phase Space ◽  
Generating Function ◽  
Series Representation ◽  
Canonical Transformations ◽  
Lie Series ◽  
Complex Phase ◽  
Lie Transformations ◽  
Complex Valued

This paper considers the Lie series representation of the canonical transformations in a complex phase space. A condition is given which selects the canonical mappings from the Lie transformations associated with a complex-valued generating function. Some special types of mappings and some simple algebraic tools are discussed.

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