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We review various semiclassical models for strong-field physics. These semiclassical models employ ensembles of classical trajectories to simulate electron motion in the continuum after being released from an atom or molecule by an external…

In this essay, we first sketch the development of ideas on the extraordinary dynamics of integrable classical nonlinear and quantum many body Hamiltonians. In particular, we comment on the state of mathematical techniques available for…

Instead of a linear system of equations for a free electromagnetic field, we propose a nonlinear system of equations. The classical electrodynamics is preseved. The appeared solutions (the electromagnetic fields) having photon properties.…

A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on…

We define functorial isomorphisms of parallel transport along etale paths for a class of G-principal bundles on a p-adic curve where G is a connected reductive algebraic group of finite presentation. This class consists of all principal…

We illustrate how classical chaotic dynamics influences the quantum properties at mesoscopic scales. As a model case we study semiclassically coherent transport through ballistic mesoscopic systems within the Landauer formalism beyond the…

The paper considers a class of linear Boltzmann transport equations which models a charged particle transport. The equation is an approximation of the original exact transport equation which involves hyper-singular integrals in their…

This paper provides some technical results needed in "Formalism for Relative Gromov-Witten Invariants." We study line-bundles on the moduli stacks of relative stable and rubber maps that are used to define relative Gromov-Witten invariants…

We describe our recent proposal of a path integral formulation of classical Hamiltonian dynamics. Which leads us here to a new attempt at hybrid dynamics, which concerns the direct coupling of classical and quantum mechanical degrees of…

It is shown that electromagnetic vortices can act as beam guides for charged particles. The confinement in the transverse directions is due to the rotation of the electric and magnetic fields around the vortex line. Large class of exact…

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in…

The effective classical/quantum dynamics of a particle constrained on a closed line embedded in a higher dimensional configuration space is analyzed. By considering explicit examples it is shown how different reduction mechanisms produce…

For electron transport in parallel-plane semiconducting structures, a model is developed that unifies ballistic and diffusive transport and thus generalizes the Drude model. The unified model is valid for arbitrary magnitude of the mean…

The problem of electron scattering on the one-dimensional complexes is considered. We propose a novel theoretical approach to solution of the transport problem for a quantum graph. In the frame of the developed approach the solution of the…

We study the quantum transport through entropic barriers induced by hardwall constrictions of hyperboloidal shape in two and three spatial dimensions. Using the separability of the Schrodinger equation and the classical equations of motion…

Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we…

We show that by integrating out the electric field and incorporating proper boundary conditions, a semiclassical Boltzmann equation can describe electron transport properties, continuously from the diffusive to ballistic regimes. General…

We present a review on the mathematical methods used to theoretically study classical propagation and quantum transport in arrays of coupled photonic waveguides. We focus on analysing two types of binary photonic lattices where…

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms and their parallel transport, that ties together various approaches. We begin with a detailed introduction to super fiber bundles. We…

We examine the spatial distribution of electrons generated by a fixed energy point source in uniform, parallel electric and magnetic fields. This problem is simple enough to permit analytic quantum and semiclassical solution, and it harbors…