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A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum o s p ( m ′ / n ′ ) ⊕ s n e ( n ′ + 1) , of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra n o ( m ′ + 1) elements times n e ( n ′ + 1) elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra n ( m ′ / n ′ ) and belong to the semidirect product Osp(m′/n′) ⋉ Ne(n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation. [ABSTRACT FROM AUTHOR]

Superlattices with coupled modes are considered to analyze entangled multipartite quantum systems. The conservation of the probability current density leads to a natural connection with Hopf fibrations, allowing the application of methods of algebraic topology to the study of electron transport. A superlattice, with n channels and d terminals, each with incoming and outgoing wave contributions, will be seen as n coupled d level quantum systems after their interaction inside a potential region. As usual, the scattering matrix connects the incoming amplitudes on the terminals with the outgoing amplitudes and probability current conservation leads to unitarity of the S-matrix and this to hyperspheres. To compute the scattering amplitudes for two colinear leads, sectionally constant potentials in the longitudinal direction and arbitrary lateral dependency are studied, which allow the analytic calculation of the amplitudes. The amplitudes are given in terms of symmetric functions of the eigenvalues by applying the Sylvester theorem for matrix functions including degenerated eigenvalues. For symplectic transfer matrices, the associated Schur functions are expressed in terms of Chebyshev polynomials in several variables, using a method developed recently, probing to be a transparent analytical and fast numerical tool. The amplitudes can then be computed for explicit superlattices, with up to two coupled modes and displayed on the associated Bloch spheres and Clifford tori, showing curves parameterized by energy and the scattering potentials. For three and four coupled modes, all necessary analytic calculations will also be reported to make possible to perform the corresponding numerical explicit evaluations in the near future. [ABSTRACT FROM AUTHOR]

In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the Poisson-Lie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass. [ABSTRACT FROM AUTHOR]

The well known Foucault nonsymmetrical pendulum is studied as a problem of sub-Riemannian geometry on nilpotent Lie groups. It is shown that in a rotating frame a sub-Riemannian structure can be naturally introduced. For small oscillations, three dimensional horizontal trajectories are computed and displayed in detail. The fiber bundle structure is explicitly shown. The underlying Lie structure is described together with the corresponding holonomy group, which turns out to be given by the center of the Heisenberg group. Other related physical problems that can be treated in a similar way are also mentioned. [ABSTRACT FROM AUTHOR]

We consider dynamical systems determined by distributions of real analytic vector fields, the equations of motion, and the associated gauge transformation are presented in detail. A gradation of the associated Lie algebras leads to the consideration of polynomial vector fields. The minimization of kinetic energy action in the class of horizontal curves associated with the distribution is formulated as a sub-Riemannian geodesic problem. Normal geodesics are fully described for the so-called Gaveau–Brockett distribution. The exponential mapping, the unitary spheres, and the wave fronts are calculated for particular cases. [ABSTRACT FROM AUTHOR]

We obtain the Lie group whose action leaves invariant the sub-Riemannian structures associated with Goursat systems and Euclidean metrics. The group naturally contains the Heisenberg group, the nilpotent group associated with the Martinet case, and the group corresponding to systems of Engel type. We compute also the Casimir functions of the associated nilpotent Poisson algebra. Our results generalize previous works on this problem of nonholonomic systems. A particular physical problem described by our model is the motion of electric charges in certain static inhomogeneous magnetic fields. We define a new algebraic curve in total space and compute two examples of sub-Riemannian extremals in cotangent space. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Bases of meromorphic vector fields on compact Riemann surfaces of genus zero and one with an arbitrary number of punctures are given. The structure constants are expressed in terms exclusively of algebraic constants. Generalized graded modules are also considered. Centrally extended algebras of vector fields as well as generalized Heisenberg algebras are constructed. The most simple examples are worked out in some detail. [ABSTRACT FROM AUTHOR]