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H. Bahlouli

Publications and source records attributed to H. Bahlouli.

At least 37 records · Page 2Linked to original sources
arXiv2017

Quantum transport across van der Waals domain walls in bilayer graphene

Bilayer graphene can exhibit deformations such that the two graphene sheets are locally detached from each other resulting in a structure consisting of domains with different inter-layer coupling. Here we investigate how the presence of these domains affect the transport properties of bilayer graphene. We derive analytical expressions for the transmission probability, and the corresponding conductance, across walls separating different inter-layer coupling domain. We find that the transmission can exhibit a valley-dependent layer asymmetry and that the domain walls have a considerable effect on the chiral tunnelling properties of the charge carriers. We show that transport measurements allow one to obtain the strength with which the two layers are coupled. We performed numerical calculations for systems with two domain walls and find that the availability of multiple transport channels in bilayer graphene modifies significantly the conductance dependence on inter-layer potential asymmetry.

cond-mat.mes-hall↗
arXiv2017

Solution of the spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using the tridiagonal representation approach

The aim of this work is to find exact solutions of the Dirac equation in 1+1 space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.

hep-th↗
arXiv2017

Energy spectrum of a generalized Scarf potential using the Asymptotic Iteration Method and the Tridiagonal Representation Approach

The well-known trigonometric Scarf potential is generalized by adding a sinusoidal term and then treated using the Asymptotic Iteration Method (AIM) and the Tridiagonal Representation Approach (TRA). The energy spectrum of the associated bound states are computed. For the AIM, we have improved convergence of the quantization condition that terminates the iterations asymptotically. This is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results (plateau of convergence). We have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point. The value of this point (or points) may depend on the physical parameters. The numerical results have been compared favorably with those resulting from the TRA.

quant-ph↗
arXiv2015

Extending the class of solutions of the Dirac equation using tridiagonal matrix representations

The aim of this work is to find exact solutions of the one-dimensional Dirac equation that do not belong to the already known conventional class. We write the spinor wavefunction as a bounded infinite sum in a complete basis set, which is chosen such that the matrix representation of the Dirac wave operator becomes tridiagonal and symmetric. This makes the wave equation equivalent to a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation and obtain the relativistic energy spectrum and corresponding state functions. Restriction to the diagonal representation results in the conventional class of solutions. As illustration, we consider the square well with half-sine potential bottom in the vector and scalar coupling under spin symmetry. We obtain the relativistic energy spectrum and show that the nonrelativistic limit agrees with results found elsewhere.

math-ph↗
arXiv2015

Interaction of solitons in dipolar Bose-Einstein condensates and formation of soliton molecules

The interaction between two bright solitons in a dipolar Bose-Einstein condensate (BEC) has been investigated aiming at finding the regimes when they form a stable bound state, known as soliton molecule. To study soliton interactions in BEC we employed a method similar to that used in experimental investigation of the interaction between solitons in optical fibers. The idea consists in creating two solitons at some spatial separation from each other at initial time $t_0$, and then measuring the distance between them at a later time $t_1 > t_0$. Depending on whether the distance between solitons has increased, decreased or remained unchanged, compared to its initial value at $t_0$, we conclude that soliton interaction was repulsive, attractive or neutral, respectively. We propose an experimentally viable method for estimating the binding energy of a soliton molecule, based on its dissociation at critical soliton velocity. Our theoretical analysis is based on the variational approach, which appears to be quite accurate in describing the properties of soliton molecules in dipolar BEC, as reflected in good agreement between the analytical and numerical results.

cond-mat.quant-gas↗
arXiv2012

The Dirac-Coulomb Problem: a mathematical revisit

We obtain a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomials techniques along with Darboux method are used to obtain the bound states energy spectrum, the relativistic scattering amplitudes and phase shifts from the asymptotic behavior of the polynomial solutions associated with the resulting three-term recursion relation.

math-ph↗
arXiv2011

Analytically Solvable 2D Potentials: Bound States

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for few exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.

quant-ph↗
arXiv2011

Analytical Treatment of the Oscillating Yukawa Potential

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix elements of the generalized Yukawa potential with complex screening parameter. This enabled us to treat analytically both the cosine and sine-like Yukawa potentials on equal footing and compute their bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we used a carefully designed complex scaling method to evaluate the resonance energies and compared our results satisfactorily with those obtained in the literature.

cond-mat.other↗
arXiv2010

Dynamical mass generation via space compactification in graphene

Fermions in a graphene sheet behave like massless particles. We show that by folding the sheet into a tube they acquire non-zero effective mass as they move along the tube axis. That is, changing the space topology of graphene from 2D to 1D (space compactification) changes the 2D massless problem into an effective massive 1D problem. The size of the resulting mass spectrum depends on the quantized azimuthal frequency and its line spacing is proportional to the inverse of the tube diameter.

cond-mat.mes-hall↗
arXiv2010

Singular Short Range Potentials in the J-Matrix Approach

We use the tools of the J-matrix method to evaluate the S-matrix and then deduce the bound and resonance states energies for singular screened Coulomb potentials, both analytic and piecewise differentiable. The J-matrix approach allows us to absorb the 1/r singularity of the potential in the reference Hamiltonian, which is then handled analytically. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian. The remaining part of the potential, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. To exhibit the power of our approach we have considered the most delicate region close to the bound-unbound transition and compared our results favorably with available numerical data.

quant-ph↗
arXiv2010

J-matrix method of scattering in one dimension: The nonrelativistic theory

We formulate a theory of nonrelativistic scattering in one dimension based on the J-matrix method. The scattering potential is assumed to have a finite range such that it is well represented by its matrix elements in a finite subset of a basis that supports a tridiagonal matrix representation for the reference wave operator. Contrary to our expectation, the 1D formulation reveals a rich and highly non-trivial structure compared to the 3D formulation. Examples are given to demonstrate the utility and accuracy of the method. It is hoped that this formulation constitutes a viable alternative to the classical treatment of 1D scattering problem and that it will help unveil new and interesting applications.

math-ph↗
arXiv2010

Extending the class of solvable potentials: III. The hyperbolic single wave

A new solvable hyperbolic single wave potential is found by expanding the regular solution of the 1D Schrödinger equation in terms of square integrable basis. The main characteristic of the basis is in supporting an infinite tridiagonal matrix representation of the wave operator. However, the eigen-energies associated with this potential cannot be obtained using traditional procedures. Hence, a new approach (the "potential parameter" approach) has been adopted for this eigenvalue problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the "parameter spectrum"). Subsequently, the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum. The bound states wavefunction is written as a convergent series involving products of the ultra-spherical Gegenbauer polynomial in space and a new polynomial in energy, which is a special case of the "dipole polynomial" of the second kind.

math-ph↗
arXiv2010

Relativistic Double Barrier Problem with Three Sub-Barrier Transmission Resonance Regions

We obtain exact scattering solutions of the Dirac equation in 1+1 dimensions for a double square barrier vector potential. The potential floor between the two barriers is higher than 2mc^2 whereas the top of the barriers is at least 2mc^2 above the floor. The relativistic version of the conventional double barrier transmission resonance is obtained for energies within + or - mc^2 from the height of the barriers. However, we also find two more (sub-barrier) transmission resonance regions below the conventional one. Both are located within the two Klein energy zones and characterized by resonances that are broader than the conventional ones.

math-ph↗
arXiv2008

The rotating Morse potential model for diatomic molecules in the J-matrix representation: II. The S-matrix approach

This is the second article in which we study the rotating Morse potential model for diatomic molecules using the tridiagonal J-matrix approach. Here, we improve further the accuracy of computing the bound states and resonance energies for this potential model from the poles of the S-matrix for arbitrary angular momentum. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian, which is included in the computations analytically without truncation. Our method has been applied to both the regular and inverted Morse potential with favorable results in comparison with available numerical data. We have also shown that the present method adds few significant digits to the accuracy obtained from the finite dimensional approach (e.g. the complex rotation method). Moreover, it allows us to handle easily both analytic and non-analytic potentials as well as 1/r singular potentials.

physics.chem-ph↗
arXiv2008

Infinite potential well with a sinusoidal bottom

We construct a tridiagonal matrix representation of the wave operator that maps the wave equation into a three-term recursion relation for the expansion coefficients of the wavefunction. Finding a solution of the recursion relation is equivalent to solving the original problem. Consequently, a larger class of solvable potentials is obtained. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. To exhibit the power of this approach, we give an exact solution for the infinite potential well with sinusoidal bottom.

quant-ph↗
arXiv2007

Solution of the wave equation in a tridiagonal representation space

We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis support an infinite tridiagonal matrix representation of the wave operator. By this requirement, we deduce a class of solvable potentials along with their corresponding bound states and stationary wavefunctions expressed as infinite series in terms of these bases. This approach allows for simultaneous treatment of the discrete (bound states) as well as the continuous (scattering states) spectrum on the same footing. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, illustrate the power of this approach in dealing with 1D quantum problems.

quant-ph↗