Journal ArticleDOI
Bernoulli sequences and trajectories in the anisotropic Kepler problem
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In this paper, the existence of at least one trajectory corresponding to each binary Bernoulli sequence is shown. But the relation between the two trajectories is not investigated, only trajectories in two dimensions with a negative energy (bound states).Abstract:
The anisotropic Kepler problem is investigated in order to establish the one‐to‐one relation between its trajectories and the binary Bernoulli sequences. The Hamiltonian has a quadratic kinetic energy with an anisotropic mass tensor and a spherically symmetric Coulomb energy. Only trajectories in two dimensions with a negative energy (bound states) are discussed. The previous study of this system was based on extensive numerical computations, but the present work uses only analytical arguments. After a review of the earlier results, their relevance to the understanding of the relation between classical and quantum mechanics is emphasized. The main new result is to show the existence of at least one trajectory corresponding to each binary Bernoulli sequence. The proof employs a number of unusual mathematical tools, although they are all elementary. In particular, the virial as a function of the momenta (rather than the action as a function of the position coordinates) plays a crucial role. Also, different ...read more
Citations
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The semiclassical evolution of wave packets
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The hydrogen atom in a uniform magnetic field — An example of chaos
TL;DR: In this paper, the hydrogen atom in a uniform magnetic field is discussed as a real and physical example of a simple nonintegrable system and the quantum mechanical spectrum shows a region of approximate separability which breaks down as we approach the classical escape threshold.
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Stochasticity in quantum systems
TL;DR: In this article, a review is devoted to an examination of quantum systems possessing stochasticity in the classical limit, and the problem of quantization of systems fulfilling the stochasticallyity condition is connected both with a wide class of problems which are physical in principle (destruction of quantum numbers due to interactions, statistics of the energy spectrum, kinetic description and so on) and with applications (molecular dynamics, interaction of atoms and molecules with a strong radiation field and such on).
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Pseudointegrable systems in classical and quantum mechanics
P.J. Richens,Michael V Berry +1 more
TL;DR: In this paper, the quantum mechanics of pseudointegrable systems are studied in detail by computing energy levels using an exact formalism, and the Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density.
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Geometric approach to Hamiltonian dynamics and statistical mechanics
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References
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Stable and Random Motions in Dynamical Systems
Jürgen Moser,Donald G. Saari +1 more
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Asymptotic solution of eigenvalue problems
Joseph B. Keller,S. I. Rubinow +1 more
TL;DR: In this article, a method for the construction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations is presented.
Book
Théorie des perturbations et méthodes asymptotiques
Victor Pavlovich Maslov,Vladimir Igorevich Arnolʹd,Vladimir Savelʹevich Buslaev,J. Lascoux,R. Sénéor,J. Leray +5 more
Journal ArticleDOI
Semiclassical calculation of bound states in a multidimensional system. Use of Poincaré’s surface of section
Donald W. Noid,Rudolph A. Marcus +1 more
TL;DR: In this article, a method utilizing integration along invariant curves on Poincare's surfaces of section is described for semiclassical calculation of eigenvalues for an anharmonically coupled pair of oscillators.