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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.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...

Based on the Sylvester and Frobenius theorems, we drastically enhance the feasibility of the transfer-matrix approach to deal with problems involving a large number of propagating and interfering modes, which require the solution of coupled differential equations and the evaluation of functions of matrix variables. We report closed formulas for the spectral decomposition of this type of functions. As specific example, besides the calculation of simple and well-known 1D one channel transfer matrices, we derive the multi-channel transfer matrix for an electron gas in the presence of a transverse electric field.

Let G be a nilpotent Lie group and let δ e lX1,X2r be a bracket generating left invariant distribution on G. In this paper we study the left invariant optimal control problem on G defined by the differential equation \begin{equation} \dot{g}(t) = u_1(t)X_1(g(t)) + u_2(t)X_2(g(t)), \end{equation} the cost functional \begin{equation} \Lambda (g,u) = \frac{1}{2}\int (u\stackrel{2}{1} + u\stackrel{2}{2})dt, \end{equation} and the family of measurable and bounded control functions t → u e (u1(t), u2(t)). We use the Pontryagin maximum principle and the associated Hamiltonian formalism to obtain the optimal controls of the system. Optimal solutions are necessarily projections of trajectories of a Hamiltonian system which in the normal case can be explicitly integrated in terms of hyperelliptic functions. Abnormal extremals (those which do not depend on the cost functional) turn out to be not strictly abnormal.

Two new analytical results are given which are of great help to understand superlattices with coupled modes. The first is an explicit relation for the transfer matrices in terms of Schur functions and Chebyshev polynomials. The second is a condition which generalizes the old well-known Floquet-Bloch trace condition to determine the spectrum. These improvements allow a fast computation of scattering amplitudes without obscuring the calculation with complicated numerical methods. As the energy grows, the eigenvalues degeneracy determines two types of transmission gaps. It is shown that these results could make it possible to design in greater detail the energy spectrum, for the very interesting case including modes coupling and degeneracy. They keep the understanding on the same footing as that of the traditional basic uncoupled problem considered at the beginning of the study of superlattices, like the Kroning-Penning model, or by Floquet-Bloch's theorem, or later by Esaki for heterostructures.

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.

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.

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.