Transfer matrix in 1D Dirac-like problems

The transfer matrix method is considered to obtain the fundamental properties of 1D Dirac-like problems. The case of 1D problems in monolayer graphene is addressed. The main characteristics of the transfer matrix are analyzed, contrasting them with the ones corresponding to 1D Schrödinger-like problems. Analytic expressions for the transmission coefficient and bound states are obtained. The continuity between bound states and states of perfect transmission is demonstrated in general, and in particular showed for the case of single electrostatic barriers. These findings in principle can be extended to 2D materials with Hamiltonian similar to monolayer graphene
such as silicene and transition metal dichalcogenides (TMDs).

Relativistic spectral bounds for the general molecular potential: application to a diatomic molecule

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 5¹Δ_g state of Na₂ molecule and compare them with the Rydberg-Klein-Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na₂(5¹Δ_g) molecule.

Advances in colloidal manipulation and transport via hydrodynamic interactions

In this review article, we highlight many recent advances in the field of micromanipulation of colloidal particles using hydrodynamic interactions (HIs), namely solvent mediated long-range interactions. At the micrsocale, the hydrodynamic laws are time reversible and the flow becomes laminar, features that allow precise manipulation and control of colloidal matter. We focus on different strategies where externally operated microstructures generate local flow fields that induce the advection and motion of the surrounding components. In addition, we review cases where the induced flow gives rise to hydrodynamic bound states that may synchronize during the process, a phenomenon essential in different systems such as those that exhibit self-assembly and swarming.

Weakly coupled bound state of 2-D Schrödinger operator with potential-measure

We consider a self-adjoint two-dimensional Schrödinger operator [Formula: see text], which corresponds to the formal differential expression[Formula: see text] where μ is a finite compactly supported positive Radon measure on [Formula: see text] from the generalized Kato class and [Formula: see text] is the coupling constant. It was proven earlier that [Formula: see text]. We show that for sufficiently small α the condition [Formula: see text] holds and that the corresponding unique eigenvalue has the asymptotic expansion[Formula: see text] with a certain constant [Formula: see text]. We also obtain a formula for the computation of [Formula: see text]. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend the results of Simon [41] to the case of potentials-measures. Also for regular potentials our results are partially new.