arXiv subjects

H. Najar

Publications and source records attributed to H. Najar.

6 recordsLinked to original sources

arXiv2026

Spectral Analysis of a Quantum Waveguide with Elliptical Window

We investigate the Dirichlet Laplacian in two spatial waveguides coupled through an elliptic window. The elliptic geometry breaks rotational symmetry and introduces anisotropy through the semi-axes of the aperture, which modifies the coupling of transverse modes and the low-lying spectrum. We prove that the operator has a finite number of discrete eigenvalues below the threshold of the essential spectrum and study their dependence on the geometric parameters of the ellipse. In contrast to the circular case, the elliptic setting gives rise to spectral effects such as eigenvalue splitting. Numerical simulations illustrate the variation of the first eigenvalues and the ground state with the window geometry.

math-ph↗

arXiv2018

A remark on the characterization of triangulated graphs

In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on the so called perfectly nested sequences.

math.CO↗

arXiv2017

Self-adjointness and spectrum of Stark operators on finite intervals

In this paper, we study self-adjointness and spectrum of operators of the form $$H=\displaystyle -\frac{d^2}{dx^2}+Fx, F>0 \quad\text{on} \quad \mathcal{H}=L^{2}(-L,L).$$ $H$ is called Stark operator and describes a quantum particle in a quantum asymmetric well. Most of known results on mathematical physics does not take in consideration the self-adjointness and the operating domains of such operators. We focus on this point and give the parametrization of all self-adjoint extensions. This relates on self-adjoint domains of singular symmetric differential operators. For some of these extensions, we numerically, give the spectral properties of $H$. One of these examples performs the interesting phenomenon of splitting of degenerate eigenvalues. This is done using the a combination of the Bisection and Newton methods with a numerical accuracy less than $10^{-8}$.

math-ph↗

arXiv2016

A quantum waveguide with Aharonov Bohm magnetic field

In a previous study \cite{n} we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$. We impose the Neumann boundary condition on a disc window of radius $a$ and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any $a>0$. In the present work we study the effect of a magnetic filed of Aharonov-Bohm type when the magnetic field is turned on this system. Precisely we prove that in the presence of such magnetic filed there is some critical values of $a_0>0$, for which we have absence of the discrete spectrum for $\displaystyle 0<\frac{a}{d}<a_0$. We give a sufficient condition for the existence of discrete eigenvalues.

math-ph↗

arXiv2009

On the discrete spectrum of a spatial quantum waveguide with a disc window

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$. We impose the Neumann boundary condition on a disc window of radius $a$ and Dirichlet boundary conditions on the remained part of the boundary of the strip. We prove that such system exhibits discrete eigenvalues below the essential spectrum for any $a>0$. We give also a numeric estimation of the number of discrete eigenvalue as a function of $\displaystyle \frac{a}{d}$. When $a$ tends to the infinity, the asymptotic of the eigenvalue is given.

math.SP↗

arXiv2008

On the singular spectrum for adiabatic quasi-periodic Schrödinger Operators

In this paper we study spectral properties of a family of quasi-periodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp in \cite{FEKL1}.

math-ph↗