Guidance, flight mechanics and trajectory optimization. Volume 12 - Relative motion, guidance equations for terminal rendezvous

Equations of relative motion and guidance for orbital transfer and docking maneuvers in spacecraft rendezvou

Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable

In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere PT invariance the superposition V(x) = x^2+ Ze^2/x of the harmonic and Coulomb potentials is defined at the purely imaginary effective charges (Ze^2=if) and regularized by a purely imaginary shift of x. This model is quasi-exactly solvable: We show that at each excited, (N+1)-st harmonic-oscillator energy E=2N+3 there exists not only the well known harmonic oscillator bound state (at the vanishing charge f=0) but also a normalizable (N+1)-plet of the further elementary Sturmian eigenstates \psi_n(x) at eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest multiplicities N we recommend their perturbative construction.Comment: 13 pages, Latex file, to appear in J. Phys. A: Math. Ge

Spin switching via quantum dot spin valves

We develop a theory for spin transport and magnetization dynamics in a quantum-dot spin valve, i.e., two magnetic reservoirs coupled to a quantum dot. Our theory is able to take into account effects of strong correlations. We demonstrate that, as a result of these strong correlations, the dot gate voltage enables control over the current-induced torques on the magnets, and, in particular, enables voltage-controlled magnetic switching. The electrical resistance of the structure can be used to read out the magnetic state. Our model may be realized by a number of experimental systems, including magnetic scanning-tunneling microscope tips and artificial quantum dot systems

PT-symmetry breaking in complex nonlinear wave equations and their deformations

We investigate complex versions of the Korteweg-deVries equations and an Ito type nonlinear system with two coupled nonlinear fields. We systematically construct rational, trigonometric/hyperbolic, elliptic and soliton solutions for these models and focus in particular on physically feasible systems, that is those with real energies. The reality of the energy is usually attributed to different realisations of an antilinear symmetry, as for instance PT-symmetry. It is shown that the symmetry can be spontaneously broken in two alternative ways either by specific choices of the domain or by manipulating the parameters in the solutions of the model, thus leading to complex energies. Surprisingly the reality of the energies can be regained in some cases by a further breaking of the symmetry on the level of the Hamiltonian. In many examples some of the fixed points in the complex solution for the field undergo a Hopf bifurcation in the PT-symmetry breaking process. By employing several different variants of the symmetries we propose many classes of new invariant extensions of these models and study their properties. The reduction of some of these models yields complex quantum mechanical models previously studied.Comment: 50 pages, 39 figures (compressed in order to comply with arXiv policy; higher resolutions maybe obtained from the authors upon request

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Quasi-Local Formulation of Non-Abelian Finite-Element Gauge Theory

Recently it was shown how to formulate the finite-element equations of motion of a non-Abelian gauge theory, by gauging the free lattice difference equations, and simultaneously determining the form of the gauge transformations. In particular, the gauge-covariant field strength was explicitly constructed, locally, in terms of a path ordered product of exponentials (link operators). On the other hand, the Dirac and Yang-Mills equations were nonlocal, involving sums over the entire prior lattice. Earlier, Matsuyama had proposed a local Dirac equation constructed from just the above-mentioned link operators. Here, we show how his scheme, which is closely related to our earlier one, can be implemented for a non-Abelian gauge theory. Although both Dirac and Yang-Mills equations are now local, the field strength is not. The technique is illustrated with a direct calculation of the current anomalies in two and four space-time dimensions. Unfortunately, unlike the original finite-element proposal, this scheme is in general nonunitary.Comment: 19 pages, REVTeX, no figure

New Quasi-Exactly Solvable Sextic Polynomial Potentials

A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial potentials is constructed. These new Hamiltonians are different from the sextic QES Hamiltonians in the literature because their eigenfunctions obey PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians present a novel problem that is not encountered when the Hamiltonian is Hermitian: It is necessary to distinguish between the parametric region of unbroken PT symmetry, in which all of the eigenvalues are real, and the region of broken PT symmetry, in which some of the eigenvalues are complex. The precise location of the boundary between these two regions is determined numerically using extrapolation techniques and analytically using WKB analysis

Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian $H_{\rm PU}$ of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that $H_{\rm PU}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.Comment: 39 pages, 0 figure

PT-Symmetric Representations of Fermionic Algebras

A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum mechanics from bosonic systems (systems for which $T^2=1$) to fermionic systems (systems for which $T^2=-1$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT-symmetric matrix representations for operator algebras of the form $\eta^2=0$, $\bar{\eta}^2=0$, $\eta\bar{\eta}+\bar {\eta} =\alpha 1$, where $\bar{eta}=\eta^{PT} =PT \eta T^{-1}P^{-1}$. It is easy to construct matrix representations for the Grassmann algebra ($\alpha=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\alpha\neq0$) if $\alpha= -1$; a matrix representation does not exist for the conventional value $\alpha=1$.Comment: 5 pages, 2 figure

Asymptotics of Expansion of the Evolution Operator Kernel in Powers of Time Interval Δt\Delta t

The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval \Dt was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as $n!$. But increasing may be more slow if the contributions with opposite signs cancel each other. Particularly, it is not excluded that for number of the potentials the expansion is convergent. For the polynomial potentials \Dt-expansion is certainly asymptotic one. The coefficients increase in this case as $\Gamma(n \frac{L-2}{L+2})$, where $L$ is the order of the polynom. It means that the point \Dt=0 is singular point of the kernel.Comment: 12 pp., LaTe