On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

A k·p Effective Hamiltonian Generator

A k·p effective Hamiltonian is important for theoretical analysis in condensed matter physics. Based on the kdotp-symmetry package, we develop an upgraded package named as kdotp-generator. This generator takes in arbitrary magnetic symmetries with their representations and returns symmetry-allowed k·p Hamiltonians. Using this package, we calculate k·p Hamiltonians for irreducible co-representations in 1651 magnetic space groups up to the third order, and their linear coupling to external fields including the electromagnetic field and the strain tensor. We hope that the package will facilitate related research in the future.     

Algebraic Hamiltonian for Vibrational Spectra of Stibine

An algebraic Hamiltonian, which in a limit can be reduced to an extended local mode model by Law and Duncan, is proposed to describe both stretching and bending vibrational energy levels of polyatomic molecules, where Fermi resonances between the stretches and the bends are considered. The Hamiltonian is used to study the vibrational spectra of stibine (SbH3). A comparison with the extended local mode model is made. Results of fitting the experimental data show that the algebraic Hamiltonian reproduces the observed values better than the extended local mode model.     

Hamiltonian Chaos in Homogeneous ADM Cosmologies?

Chaos in dynamical systems has best been understood in terms of Hamiltonian systems. A primary method of diagnosis of chaos in these systems is the Lyapunov exponent. According to general relativity, space-time is itself a dynamical system. When the evolution of a model universe is expressed in the ADM form it can be described as a Hamiltonian system. Among the various model cosmologies, the Mixmaster or Bianchi IX cosmology has been extensively studied as a candidate to exhibit chaos. However, the Lyapunov exponents in this system have shown contradictory properties, including a seeming dependence on the coordinates used to describe space-time. Such dependencies, if true, would be surprising as the time coordinate of space-time is unrelated to the parameterization of phase space. Further, this sort of dependence would relegate chaos to a bad coordinate choice rather than a dynamic property of the system. The problem with the Lyapunov exponent lies in the ambiguities remaining in the ADM action integral. The current interpretation involves an arbitrary Lagrange multiplier—thought to be necessary for the coordinate invariance of space-time. An arbitrary multiplier turns out to be unnecessary for coordinate invariance, and in addition destroys the symplectic structure of phase space. In reality, the geometry selects the parameterization of phase space, and any change in the parameter results in a changed Hamiltonian system. It must be emphasized that the fixing of the phase space parameter does NOT impose a coordinate choice on space-time. The parameter is selected by the symplectic structure of phase space and full coordinate invariance of space-time is left intact. Once the demands of both geometries, space-time and phase space, have been satisfied, the Lyapunov exponent becomes independent of the coordinate imposed on space-time. Additionally, the correction of the phase space structure leads to a Hamiltonian that is more general, in that it describes a gravitational system with a cosmological constant, than is currently the case.     

Revivals in Caldeira–Leggett Hamiltonian dynamics

We reconsider the problem of quantum system interacting with a complex environment discussed by Caldeira and Leggett (CL), and generalize their results for a quantum oscillator coupled to a reservoir R with dense discrete spectrum of oscillators with close to ${\omega }_s$ frequencies. Dynamics consists of recurrence cycles with partial revivals of the initial state. This revival or Loschmidt echo appears in each cycle. Width and number of the Loschmidt echo components increase with the recurrence cycle number leading to irregular, stochastic-like time evolution.     

Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of to include any Hamiltonian for which a certain non-resonance condition holds. Received: 5 October 1999 / Accepted: 2 February 2000     

Unbounded Trace Orbits of Thue–Morse Hamiltonian

It is well known that, an energy is in the spectrum of Fibonacci Hamiltonian if and only if the corresponding trace orbit is bounded. However, it is not known whether the same result holds for the Thue–Morse Hamiltonian. In this paper, we give a negative answer to this question. More precisely, we construct two subsets \(\Sigma _{II}\) and \(\Sigma _{III}\) of the spectrum of the Thue–Morse Hamiltonian, both of which are dense and uncountable, such that each energy in \(\Sigma _{II}\cup \Sigma _{III}\) corresponds to an unbounded trace orbit. Exact estimates on the norm of the transfer matrices are also obtained for these energies: for \(E\in \Sigma _{II}\cup \Sigma _{III}, \) the norms of the transfer matrices behave like

$$\begin{aligned} e^{c_1\gamma \sqrt{n}}\le \Vert T_{ n}(E)\Vert \le e^{c_2\gamma \sqrt{n}}. \end{aligned}$$

However, two types of energies are quite different in the sense that each energy in \(\Sigma _{II}\) is associated with a two-sided pseudo-localized state, while each energy in \(\Sigma _{III}\) is associated with a one-sided pseudo-localized state. The difference is also reflected by the local dimensions of the spectral measure: the local dimension is 0 for energies in \(\Sigma _{II}\) and is larger than 1 for energies in \(\Sigma _{III}.\) As a comparison, we mention another known countable dense subset \(\Sigma _I\). Each energy in \(\Sigma _I\) corresponds to an eventually constant trace map and the associated eigenvector is an extended state. In summary, the Thue–Morse Hamiltonian exhibits “mixed spectral nature”.

     

The Hamiltonian of Asymptotically Friedmann-Lemaître-Robertson-Walker Spacetimes

We obtain the correct Hamiltonian which describes the dynamics of classes of asymptotic open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, which includes Tolman geometries. We calculate the surface term that has to be added to the usual Hamiltonian of General Relativity in order to obtain an improved Hamiltonian with well defined functional derivatives. For asymptotic flat FLRW spaces, this surface term is zero, but for asymptotic negative curvature FLRW spaces it is not null in general. In the particular case of the Tolman geometries, they vanish. The surface term evaluated on a particular solution of Einstein's equations may be viewed as the energy of this solution with respect to the FLRW spacetime they approach asymptotically. Our results are obtained for a matter content described by a dust fluid, but they are valid for any perfect fluid, including the cosmological constant.     

Temperature effects of Hamiltonian dark solitons on the stability of holographic solitons in a separate holographic–Hamiltonian soliton pair

We investigate theoretically the temperature effects on the evolution and stability of a separate holographic–Hamiltonian dark–dark or bright–dark soliton pair formed in an unbiased serial photorefractive crystal circuit. Our numerical results show that, for a stable dark–dark or bright–dark soliton pair originally formed in a crystal circuit at given temperatures, when the crystal in which formed a Hamiltonian dark soliton changes, the holographic dark or bright soliton supported by the other crystal tends to evolve into another stable soliton or experiences larger cycles of compression or breaks up into beam filaments or exhibit a common decaying process. The holographic dark soliton is more sensitive to the temperature change than the holographic bright one.     

磁性物质交换Hamiltonian中两项的竞争

对于一个N电子体系, 正确的交换Hamilton应该由两项组成，为H_ex=-2A1ii·s_j-2A₂ii·sj，而不是以往的铁磁学理论使用的H_ex=-2Aii ·s_j （其中A为A₁与A₂的代数和， A₁>0, A₂<0）, 以往的理论使用了一个不合理的交换Hamiltonian量.-2A₁ii·s_j与-2A₂ii< /sub>·s_j在数学上是同类项，但是在物理上不是 同类项，它们有不同的本征态和本征值.根据量子力学中的态叠加原理，这个电子系统的本 征态矢为X〉=1A²₁+A²₂(A₁ 1〉+A₂‖2〉),其中Dirac符号1〉表示系统所有电子 的自旋平行排列时的态（简称平行自旋态）矢量，2〉表示系统所有电子或最近邻电子的自 旋反平行排列时的态（简称反平行自旋态）矢量，H_ex的本征值（即系统的 交换能） 为E=-Nz（A₁-A₂）-2NzA₂²A₁ +A₂=-Nz（A₂-A₁）-2NzA²₁A< sub>1+A₂,其中z为最近邻电子数.当A₂=0时，X〉=1〉，E =-A₁, 系统具有Wei ss 铁磁性；当A₁ =0 时，X〉=2〉，E =-A₂，系统具有Neel 反铁磁性；当A₁ =A₂（即A=0）时，X〉=12 (1〉+2〉)，E=-A₁，系统处于自旋玻璃（spin glass）态；当A₁>A ₂时，X〉=1A²₁+A²₂［(A₁-A₂)1〉+A₂(1〉+2〉)］,平行自旋态与自旋 玻璃态共存；当A₁2时，X〉=1A²₁+A2₂［(A₂-A₁)2〉+A₁( 1〉+2〉)］，反平行自旋态与自旋玻 璃态共存.与原来理论中的Weiss铁磁态或Neel反铁磁态相比，平行自旋态与自旋玻璃态共存 或反平行自旋态与自旋玻璃态共存使系统的交换能降低.自旋玻璃态中电子自旋之间取向的 随机性或无序性是由交换Hamiltonian中-2A₁iis_j与-2A₂ii·s_j之间的竞争引起的，不是热运 动引起的. 关键词： 交换哈密顿量 铁磁态 反铁磁态 自旋玻璃态     

Hamiltonian and Thermodynamic Modeling of Quantum Turbulence

The state variables in the novel model introduced in this paper are the fields playing this role in the classical Landau-Tisza model and additional fields of mass, entropy (or temperature), superfluid velocity, and gradient of the superfluid velocity, all depending on the position vector and another tree dimensional vector labeling the scale, describing the small-scale structure developed in ⁴He superfluid experiencing turbulent motion. The fluxes of mass, momentum, energy, and entropy in the position space as well as the fluxes of energy and entropy in scales, appear in the time evolution equations as explicit functions of the state variables and of their conjugates. The fundamental thermodynamic relation relating the fields to their conjugates is left in this paper undetermined. The GENERIC structure of the equations serves two purposes: (i) it guarantees that solutions to the governing equations, independently of the choice of the fundamental thermodynamic relation, agree with the observed compatibility with thermodynamics, and (ii) it is used as a guide in the construction of the novel model.     

Measure Synchronization of High-Cycle Islets in Coupled Hamiltonian Systems

Measure synchronization is a new phenomenon found in coupled Hamiltonian systems recently and it is interesting to understand its properties comprehensively. We discuss the measure synchronization of a coupled pair of standard maps in high period quasi-period orbits, and the measure synchronization transition is associated with the transition of coupled systems from quasi-periodicity to chaos. This behaviour is very different from that found by Hampton and Zanette [Phys. Rev. Lett. 83 (1999) 2179].     

Solitonic eigenstates of the chaotic Bose–Hubbard Hamiltonian

We study the evolving energy spectrum of interacting ultra-cold atoms in an optical lattice as a function of an external parameter, the tilt of the lattice. In a regime where the quantum mechanical model, the Bose–Hubbard Hamiltonian, shows predominantly chaotic behavior, we identify regular structures in the parametric level evolution and characterize the eigenstates associated with these structures. The mechanism generating these structures is found to be different from Stark localization or energetic isolation and is induced by an interplay of driving and interaction.     

Geometric phase of a classical Aharonov–Bohm Hamiltonian

We present a gauge-invariant approach for associating a geometric phase with the phase space trajectory of a classical dynamical system. As an application, we consider the classical analog of the quantum Aharonov–Bohm (AB) Hamiltonian for a charged particle orbiting around a current carrying long thin solenoid. We compute the classical geometric phase of a closed phase space trajectory, and also determine its dependence on the magnetic flux enclosed by the orbit. We study the similarities and differences between this classical geometric phase and the AB phase acquired by the wave function of the quantum AB Hamiltonian. We suggest an experiment to measure the geometric phase for the classical AB system, by using an appropriate optical fiber ring interferometer.     

Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values

Hamiltonian formalism of the mKdV equation with non-vanishing boundary value is re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 （1988） 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.