arXiv subjects

Hassan Babaei

Publications and source records attributed to Hassan Babaei.

4 recordsLinked to original sources

arXiv2026

Rigorous analysis for the Dirac system on the quarter-plane

Considered and analyzed below are fully non-homogeneous initial-boundary-value problems for the celebrated Dirac system, formulated on the spatial half-line. Analytical solution formulae are derived formally via suitable implementation of the well-known Fokas' unified transform methodology, and rigorously verified a posteriori. The latter substantial task relies on complex-analytic tools and careful interpretation of the obtained integral representations. These valid solutions are then used for investigating qualitative properties. These include boundary behavior near the axes of the domain as well as long-range asymptotics and long-time (eventual) periodicity. Notably, smoothness of the solution, both within and upto the boundary of the domain, depends heavily on certain compatibility conditions between initial, boundary and forcing data. Further results pertaining to solution's regularity and uniqueness are thence established based on the qualitative theory. The closed-form expressions reported here are also useful in the study of non-linear counterparts.

math.AP↗

arXiv2026

Determining wavenumbers for Hall and electron magnetohydrodynamics turbulence

In turbulent flows, the Kolmogorov wavenumber characterizes the smallest scales at which viscous effects dominate. A mathematical analogue of this notion first introduced by Foias and Prodi [8] -- a determining wavenumber -- quantifies the minimal set of modes that uniquely determine the long-time behavior of solutions. Extending this framework from the Navier-Stokes equations to magnetized plasma models, we focus on the Hall-MHD and Electron-MHD turbulence in sub-ion and dissipation ranges. We prove existence of time-dependent determining wavenumbers for weak solutions of the Hall- and electron-MHD, improving upon previous results that were not optimal and lacked any comparison with phenomenological dissipation scales. Under explicit scale-localized intermittency assumptions, we show that their time averages are bounded above by Kolmogorov-like dissipation wavenumbers predicted by phenomenological studies of plasma turbulence. For strong electron-MHD solutions, we also establish a uniform bound on the magnetic determining wavenumber from Besov regularity.

math.AP↗

arXiv2025

Well-posedness of the electron MHD with partial resistivity

Due to the singular nonlinear Hall term, the non-resistive electron magnetohydrodynamics (MHD) is not known to be locally well-posed in general. In this paper we consider the $2\frac12$D electron MHD with either horizontal or vertical resistivity and show local well-posedness in Sobolev spaces.

math.AP↗

arXiv2017

Quantum Mechanics of a Photon

A first quantized free photon is a complex massless vector field $A=(A^μ)$ whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space $\mathscr{H}$ of the photon by endowing the vector space of the fields $A$ in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in $\mathscr{H}$, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis, and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.

quant-ph↗