Fractional Zero Curvature Equation and Generalized Hamiltonian Structure of Soliton Equation Hierarchy

We presented the fractional zero curvature equation and generalized Hamiltonian structure by using of the differential forms of fractional orders. Example of the fractional AKNS soliton equation hierarchy and its Hamiltonian system are obtained.     

Noether symmetry for fractional Hamiltonian system

Fractional Hamiltonian systems within combined Riemann-Liouville fractional order derivative and combined Caputo fractional order derivative are established. Then Noether quasi-symmetry and conserved quantity for the fractional Hamiltonian systems are presented. Thirdly, perturbation to Noether quasi-symmetry and adiabatic invariant are studied. Several special cases are discussed in each section. And finally, two applications, i.e., the fractional Lotka biochemical oscillator model and the fractional isotropic harmonic oscillator model, are discussed to illustrate the results and methods.     

Theory of four-dimensional fractional quantum Hall states

We propose a pseudo-potential Hamiltonian for the Zhang-Hu’s generalized fractional quantum Hall states to be the exact and unique ground states. Analogously to Laughlin’s quasi-hole (quasi-particle), the excitations in the generalized fractional quantum Hall states are extended objects. They are vortex-like excitations with fractional charges +(−)1/m³ in the total configuration space CP³. The density correlation function of the Zhang-Hu states indicates that they are incompressible liquid.     

Fractional Hamilton’s equations of motion in fractional time

The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.      

Singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.     

Adiabatic switching approach to multidimensional supersymmetric quantum mechanics for several excited states

We present a time-dependent method for determining several approximate excited-state energies and wave functions using a vectorial approach to multidimensional supersymmetric quantum mechanics. First, a vectorial approach is used to generate the tensor sector two Hamiltonian, which is isospectral with the original scalar sector one Hamiltonian above the ground state of the sector one Hamiltonian. We construct a time-dependent Hamiltonian interpolating between the scalar sector one Hamiltonian and the tensor sector two Hamiltonian. Then, we can adiabatically switch from the ground state of the sector one Hamiltonian to the ground state of the sector two Hamiltonian by solving the time-dependent Schrödinger equation. In addition, by employing an initial wave packet orthogonal to that leading to the ground state of sector two, we also obtain the first-excited state of sector two. Construction of the orthogonal sector one states is trivial due to the tensor nature of sector two. The ground and first-excited states of the sector two Hamiltonian can be used with the charge operator to obtain the first two excited state wave functions of the sector one Hamiltonian. Excellent computational results are obtained for two-dimensional nonseparable degenerate and nondegenerate systems.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

On a coupled random walk process specified by a Hamiltonian

A coupled random walk process specified by an effective Hamiltonian in a potential field is proposed. The Hamiltonian is expressed in terms of a set of jumping probabilities which characterize the random walk processes. The steady state is expressed by the Hamiltonian. Conditions for the Hamiltonian to be reduced to the Ginzburg-Landau type are discussed.     

Experimental Hamiltonian Learning of an 11-Qubit Solid-State Quantum Spin Register

Learning the Hamiltonian of a quantum system is indispensable for prediction of the system dynamics and realization of high fidelity quantum gates.However,it is a significant challenge to efficiently characterize the Hamiltonian which has a Hilbert space dimension exponentially growing with the system size.Here,we develop and implement an adaptive method to learn the effective Hamiltonian of an 11-qubit quantum system consisting of one electron spin and ten nuclear spins associated with a single nitrogen-vacancy center in a diamond.We validate the estimated Hamiltonian by designing universal quantum gates based on the learnt Hamiltonian and implementing these gates in the experiment.Our experimental result demonstrates a well-characterized 11-qubit quantum spin register with the ability to test quantum algorithms,and shows our Hamiltonian learning method as a useful tool for characterizing the Hamiltonian of the nodes in a quantum network with solid-state spin qubits.     

An algorithm for obtaining an optimalized projected Hamiltonian and its groundstate

An algorithm is proposed for solving the problem of projecting the Hamiltonian onto a finite dimensional subspace in such a way that the groundstate of the projected Hamiltonian represents an optimalized approximation to the groundstate of the full (unprojected) Hamiltonian and simultaneously for obtaining the groundstate of the projected Hamiltonian.     

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.     

应用矩阵函数理论解薜定格方程

本文提出应用矩阵函数理论来解薛定格方程的方法。讨论了哈密顿量不显含t和显含t二种情况,对于哈密顿量不显含t的情况,矩阵函数理论提供了完全规范化的解法,对于哈密顿量显含t的情况,只对二能级系统作了讨论。     

An exactly solvable model for the large polaron

We present an exactly diagonalizable model Hamiltonian for the large polaron derived by analyzing the variational ansatz by Haga-Larsen (HL) for the Fröhlich Hamiltonian. The lowest energy eigenvalue of the model Hamiltonian for fixed wave numbers reproduces the energy of the variational ansatz by Haga-Larsen and is, therefore, an upper bound with respect to the corresponding energy eigenvalue of the Fröhlich Hamiltonian. This is valid for any momentum which is proven by extending the Haga-Larsen approach. Furthermore, since all integrations can be performed analytically, the model Hamiltonian is easily tractable. The energy eigenvalue spectrum of the model Hamiltonian is studied below and above the phonon-emission threshold. The quality of the model Hamiltonian is determined by the variational ansatz of Haga and Larsen. Incorporating an improved energy-momentum relation, a generalized model Hamiltonian is derived possessing a larger validity range with respect to the coupling strength. Furthermore, a second exactly diagonalizable model Hamiltonian based on improved Wigner-Brillouin perturbation theory due to Warmenbol, Peeters, and Devreese (WPD) is presented. It is briefly demonstrated that one is able to construct all mentioned model Hamiltonians also in the 2D polaron problem. In contrast to the 3D case, where the HL-type model Hamiltonian possesses the higher quality for any momentum, in the 2D case, it works well only for small momenta. For large momenta, only the WPD-type model Hamiltonian describes the energy-momentum relation correctly. We demonstrate the usefulness of the model Hamiltonian concept by exactly calculating the one-electron Green’s function for all mentioned model Hamiltonians and comment why significant advantages of the model Hamilton concept for the treating of low-dimensional systems (planar semiconducting quantum-well structures) can be expected.     

A Non-unitary Mapping from Cooper Pairs to Bosons

We derive a non-hermitian boson-fermion Hamiltonian, that is equivalent to the entirely fermionic Richardson Hamiltonian which describes the dynamics of conduction electrons in a superconductor. This is done using a generalized Dyson mapping, that replaces Cooper pairs with bosons. We show that the calculation of some physical quantities is simpler when one uses the boson-fermion Hamiltonian rather than the original Richardson Hamiltonian.     

Comparison of several approaches to the relativistic dynamics of directly interacting particles

Several approaches to the relativistic dynamics of directly interacting particles are compared. The equivalence between constrained Hamiltonian relativistic systems and a priori Hamiltonian predictive ones is completely proved. Coordinate transformations are obtained to express these systems in the framework of noncovariant predictive mechanics. The world line condition for constrained Hamiltonian relativistic systems is analyzed and is proved to be also necessary in the predictive Hamiltonian framework.     

Fractional hamilton formalism within caputo’s derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.     

键长-键角和Radau坐标下哈密顿算符在傅里叶基组表象下的厄米性

在量子动力学计算中,有时候为了规避奇点问题或者节省计算量,我们经常需要对哈密顿量进行变换. 然而,在使用傅里叶基矢计算时,哈密顿量的变换形式容易导致哈密顿矩阵失去厄米性,进而有些情况下使数值计算变得不稳定. 本文主要讨论构建具有厄米性的哈密顿算符的方法. 以三原子分子为例,构建了键长—键角和Radau坐标下描述分子运动的各种形式的哈密顿量. 基于这些哈密顿量,采用含时波包方法计算了OClO分子的吸收光谱,讨论了非厄米性矩阵对计算结果的影响. 本文所得到的结论对基于基函数展开的量子动力学计算都是适用的.     

Invariant combinations of coefficients in the rotational Hamiltonian

A general development of combinations of parameters in the asymmetric rotor Hamiltonian, which are independent of the coefficients of the contact transformation operators used in the reduction of that Hamiltonian, is given. These invariant combinations are obtained for a Hamiltonian containing up to eighth degree operators. The adaptation to the A and S forms of the Watson Hamiltonian and their utility in conversion of coefficients from one form to another is discussed.     

Epicycloids generating Hamiltonian minimal surfaces in the complex quadric

We show that Hopf tubes on Lancret curves shaped over an epicycloid are Hamiltonian minimal surfaces in the complex quadric. Moreover they are the only Hopf tubes that are Hamiltonian minimal there. This allows one to connect two apparently unrelated topics, such as Hamiltonian minimal surfaces and curves with constant precession, and more generally slant helices. Furthermore, Hamiltonian minimal Hopf tubes encode the phases of particles described according to the gyroscopic force theory.     

Constructing the time independent Hamiltonian from a time dependent one

In this paper we introduce a method for finding a time independent Hamiltonian of a given Hamiltonian dynamical system by canonoid transformation of canonical momenta. We find a condition that the system should satisfy to have an equivalent time independent formulation. We study the example of a damped harmonic oscillator and give the new time independent Hamiltonian for it, which has the property of tending to the standard Hamiltonian of the harmonic oscillator as damping goes to zero.