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A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum [Formula: see text], 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 [Formula: see text] elements times [Formula: see text] 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 [Formula: see text] and belong to the semidirect product Osp( m′/ n′) ⋉ N e( 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.

Time dependent quantum problems defined by quadratic Hamiltonians are solved using canonical transformations. The Green’s function is obtained and a comparison with the classical Hamilton–Jacobi method leads to important geometrical insights like exterior differential systems, Monge cones and time dependent Gaussian metrics. The Wei–Norman approach is applied using unitary transformations defined in terms of generators of the associated Lie groups, here the semi-direct product of the Heisenberg group and the symplectic group. A new explicit relation for the unitary transformations is given in terms of a finite product of elementary transformations. The sequential application of adequate sets of unitary transformations leads naturally to a new separation of variables method for time dependent Hamiltonians, which is shown to be related to the Inonu–Wigner contraction of Lie groups. The new method allows also a better understanding of interacting particles or coupled modes and opens an alternative way to analyze topological phases in driven systems.

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp . The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

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...

Probability electronic current conservation in superlattices leads to a natural connection with the qubit and quantum gate concepts of quantum information and the introduction of Bloch spheres and Hopf bundles. Superlattices or multilayer devices with coupled channels are described by sectionally constant potentials in the longitudinal direction and arbitrary lateral dependency, which allow the explicit analytic calculation of the scattering amplitudes connecting ingoing with outgoing wave functions. A superlattice with n open channels and two terminals can be seen as a quantum gate, since both can be viewed as quantum systems with n components, each of which can have two states given by input and output. Taking into account the dimensionality of the respective Hilbert spaces, it results that the one-channel or mode superlattice corresponds to a single qubit; the two-channels case to two qubits, and the three- and four-channels cases to three qubits. The coupling of modes or channels corresponds to the entanglement of qubits. As shown in this work, superlattices with interacting energy modes constitute physical systems which allow to understand and evaluate explicitly the main features used to describe entangled qubits through the study of Hopf fiber bundles on spheres. Explicit examples for superlattice gate models with coupled channels are provided.

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.

We study the sub-Riemannian structure determined by a left-invariant distribution of rank n on a step-2 simply-connected nilpotent Lie group G of dimension n(n+1)/2. We describe a transitive group action that leaves invariant the sub-Riemannian structure. By using geometric optimal control theory techniques, we derive necessary conditions for the length-minimality of the sub-Riemannian geodesics. We perform an integration process for the associated Hamiltonian system that yields explicit expressions for the extremal curves and the corresponding sub-Riemannian geodesics, the obtained formulas allow the complete parametrization of the exponential mapping in terms of algebraic invariants of the problem.

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.

Let H denote either the Heisenberg group \Bbb R^{2n+1}, or the Cartesian product of n copies of the three-dimensional Heisenberg group \Bbb R^3. Let lX1, Y1, …, Xn, Ynr be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics \dot{q}(t)=\sum\limits^n_{i=1}u_i(t)X_i(q(t))+v_i(t)Y(q(t)), the cost functional \Lambda(q,u)={1 \over 2}\int \sum\limits^n_{i=1}\mu_i(u^2_i+v^2_i), with arbitrary positive parameters μ1, …, μn, and admissible controls taken from the set of measurable functions t\mapsto u(t)=(u_1(t),v_1(t),\ldots , u_n(t), v_n(t)). The above control system is encoded either in the kernel of a contact 1-form (for \Bbb R^{2n+1}), or in the kernel of a Pfaffian system (for \Bbb R^{3n}). In both cases, the action of the semi-direct product of the torus Tn with H describe the symmetries of the problem. The Pontryagin maximum principle provides optimal controlss extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant. An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.

The total nuclear level density is obtained for periodic single particle spectra. First, we express the canonical partition function in terms of Jacobi theta functions for degenerated fermion systems. Secondly, we use a relation for the number of representations of an integer number as sums of smaller numbers to find an exact formula for the nuclear level density. The shell structure is incorporated in the result in a simple way. An asymptotic relation can be easily obtained from the exact result and shown to be the well known back-shifted Bethe-formula.

We show here that a transverse electric field, as in field-effect devices, leads to interesting interfering phenomena in the electronic transport. Using the Sylvester Theorem for matrix-valued functions, we study the transmission properties of a multi-propagating modes (multichannel) quasi-two-dimensional electron gas in the presence of a channel mixing transverse electric field. We observe interesting resonant coupling effects, which explain the fundamentals of the giant superlattice-conductance as well as giant superlattice-resistance effects.

A new formula for the nuclear particle-hole state density is derived. Constant single particle spectra for neutrons and protons are assumed. The so-called Williams formula is extended to different numbers of particles and holes. Correction terms are also obtained. An infinite depth of the potential well is considered.

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.

A Hamiltonian approach is presented to study the two dimensional motion of damped electric charges in time dependent electromagnetic fields. The classical and the corresponding quantum mechanical problems are solved for particular cases using canonical transformations applied to Hamiltonians for a particle with variable mass. The Green's function is constructed and, from it, the motion of a Gaussian wave packet is studied in detail., Comment: 14 pages, 4 figures, misprints and figures corrected

By assuming periodic discrete electronic spectra, the properties of small metal particles are analyzed. The canonical partition function is obtained exactly. The heat capacity and the electronic magnetic susceptibility are calculated in the presence of a static magnetic field. These results are an extension of the calculations for an equally spaced spectrum. Preliminary considerations of the statistical theory of spectra are included.

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.

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.

We consider a hierarchy of non-holonomic con- straints defined by means of Pfaffian systems of differential 1-forms, that yield driftless control systems affine in the con- trols. The first level of the hierarchy produces driftless control systems defined by distributions of vectors fields determining step-2 nilpotent Lie algebras that can be studied by standard geometric control theory methods. We present the case of the six dimensional Lie algebra that models a classical particle in a three-dimensional space in a homogeneous magnetic field with arbitrary direction.

We propose a formulation for charged non relativistic particles in homogeneous magnetic field within the framework of Sub-Riemannian geometry. Our approach takes advantage of the underlying Lie algebraic structure to uncover a set of Casimir elements. Second order conditions for the corresponding variational problem are touched upon.

In this paper we study the sub-Riemannian geodesic problem determined by a left invariant distribution of dimension n in a nilpotent Lie group G of dimension n(n+1)/2. The problem is formulated as an left invariant optimal control sytem. We describe a group action that leaves invariant the system. We apply Pontryagin maximum to derive necessary conditions for the optimality of the geodesics. For n odd abnormal extremals do exists, but they turn out to be non-strictly abnormal. The adjoint equation for the normal extremals is explicitly integrated. At the end, we specialize our results in some of the known low dimensional cases, namely, the Heisenberg (2,3) case and the (3,6) case.

We consider the action of a Lie group that leaves invariant the sub-Riemannian structures associated to Goursat systems and Euclidean metrics. The low dimensional cases contain the well-known Heisenberg group, as well as the nilpotent groups associated to the Martinet and Engel systems. For arbitrary dimensions, we establish the complete integrability of the associated adjoint system by means of the explicit calculation of Casimir functions of the associated nilpotent Poisson algebra. We define a new algebraic curve in total space and compute two examples of sub-Riemannian extremals in cotangent space. Our approach fits certain non-holonomic classical problems, particularly that of the physical model for a plasma considered as a gas of non interacting charged particles under the influence of a static non homogeneous magnetic field.