Abstract:
In this paper we construct a distinguished Riemannian geometrization on the dual 1-jet space J^{1*}(T,M) for the multi-time quadratic Hamiltonian functions. Our geometrization includes a nonlinear connection N, a generalized Cartan canonical N-linear connection (together with its local d-torsions and d-curvatures), naturally provided by a given quadratic Hamiltonian function depending on polymomenta.

Abstract:
We represent a classical Maxwell-Bloch equation and related to it positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra $n_+$ of affine Lie algebra $\hat {sl}_2$ on a Maxwell-Bloch phase space treated as a homogeneous space of $n_+$. A space of local integrals of motion is described using cohomology methods. We show that hamiltonian flows associated to the Maxwell-Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an abelian subalgebra of the nilpotent subalgebra $n_+$ on a Maxwell- Bloch phase space. Possibilities of quantization and latticization of Maxwell-Bloch equation are discussed.

Abstract:
Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly-correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of p and d orbitals. This paper derives the most general s, p and d orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full configuration interaction method to study p and d orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity.

Abstract:
It is shown that the radial part of the Hydrogen Hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant epsilon. The 1-susy approach is used to construct non-hermitian Hamiltonians with hydrogen spectra. Other non-hermitian Hamiltonians are shown to admit an extra `complex energy' at epsilon. New self-adjoint hydrogen-like Hamiltonians are also derived by using a 2-susy transformation with complex conjugate pairs epsilon, (c.c) epsilon.

Abstract:
The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.

Abstract:
The equations of the linearized first post-Newtonian approximation to general relativity are often written in "gravitoelectromagnetic" Maxwell-like form, since that facilitates physical intuition. Damour, Soffel and Xu (DSX) (as a side issue in their complex but elegant papers on relativistic celestial mechanics) have expressed the first post-Newtonian approximation, including all nonlinearities, in Maxwell-like form. This paper summarizes that DSX Maxwell-like formalism (which is not easily extracted from their celestial mechanics papers), and then extends it to include the post-Newtonian (Landau-Lifshitz-based) gravitational momentum density, momentum flux (i.e. gravitational stress tensor) and law of momentum conservation in Maxwell-like form. The authors and their colleagues have found these Maxwell-like momentum tools useful for developing physical intuition into numerical-relativity simulations of compact binaries with spin.

Abstract:
We study which geometric structure can be constructed from the vierbein (frame/coframe) variables and which field models can be related to this geometry. The coframe field models, alternative to GR, are known as viable models for gravity, since they have the Schwarzschild solution. Since the local Lorentz invariance is violated, a physical interpretation of additional six degrees of freedom is required. The geometry of such models is usually given by two different connections -- the Levi-Civita symmetric and metric-compatible connection and the Weitzenbock flat connection. We construct a general family of linear connections of the same type, which includes two connections above as special limiting cases. We show that for dynamical propagation of six additional degrees of freedom it is necessary for the gauge field of infinitesimal transformations (antisymmetric tensor) to satisfy the system of two first order differential equations. This system is similar to the vacuum Maxwell system and even coincides with it on a flat manifold. The corresponding ``Maxwell-compatible connections'' are derived. Alternatively, we derive the same Maxwell-type system as a symmetry conditions of the viable models Lagrangian. Consequently we derive a nontrivial decomposition of the coframe field to the pure metric field plus a dynamical field of infinitesimal Lorentz rotations. Exact spherical symmetric solution for our dynamical field is derived. It is bounded near the Schwarzschild radius. Further off, the solution is close to the Coulomb field.

Abstract:
Previous $\lambda$-deformed {\it non-Hermitian} Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed $\lambda$-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

Abstract:
We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schroedinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painleve transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons, and Jacobi elliptic and second Painleve transcendental solutions for several variable Hamiltonians that are important for current research in nonlinear optics and Bose-Einstein condensation. The Feshbach resonance matter wave soliton management is briefly discussed from this new perspective.