scholarly journals On the fractional Dirac systems with non-singular operators

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2159-2168
Author(s):  
Ahu Ercan
Keyword(s):  
Laplace Transforms ◽  
Dirac System ◽  
Singular Operators ◽  
Dirac Systems

In this manuscript, we consider the fractional Dirac system with exponential and Mittag-Leffler kernels in Riemann-Liouville and Caputo sense. We obtain the representations of the solutions for Dirac systems by means of Laplace transforms.

Levinson's theorem and Titchmarsh-Weyl m(λ) theory for Dirac systems*

1988 ◽  
Vol 109 (1-2) ◽  
pp. 173-186 ◽  
Author(s):  
D. B. Hinton ◽  
M. Klaus ◽  
J. K. Shaw
Keyword(s):  
Asymptotic Behaviour ◽  
Bound States ◽  
Dirac System ◽  
Levinson’S Theorem ◽  
Dirac Systems ◽  
Levinson Theorem ◽  
Asymptotic Phase

SynopsisA Levinson theorem is proved for a Dirac system with one singular endpoint. The number ofbound state is expressed in terms of the change in asymptotic phase of an appropriate solution and in terms of factors whose values depend on the presence of half-bound states. The behaviour of the asymptotic phase is used to determine the asymptotic behaviour of the Titchmarsh-Weyl m-function.

scholarly journals Numerical Algorithms for Computing Eigenvalues of Discontinuous Dirac System Using Sinc-Gaussian Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. M. Tharwat ◽  
A. Al-Fhaid
Keyword(s):  
Error Analysis ◽  
Numerical Algorithms ◽  
High Accuracy ◽  
Transmission Conditions ◽  
Numerical Results ◽  
New Method ◽  
Dirac System ◽  
Sinc Method ◽  
Dirac Systems ◽  
Gaussian Method

The eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity are computed using the sinc-Gaussian method. The error analysis of this method for solving discontinuous regular Dirac system is discussed. It shows that the error decays exponentially in terms of the number of involved samples. Therefore, the accuracy of the new method is higher than the classical sinc-method. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Comparisons with the classical sinc-method are given.

scholarly journals Direct and inverse spectral problems for Dirac systems with nonlocal potentials

2019 ◽  
Vol 39 (5) ◽  
pp. 645-673
Author(s):  
Kamila Dębowska ◽  
Leonid P. Nizhnik
Keyword(s):  
Characteristic Function ◽  
Dirac System ◽  
Inverse Spectral Problems ◽  
One Dimensional ◽  
Spectral Problems ◽  
Dirac Systems ◽  
Inverse Spectral ◽  
The One ◽  
The Given ◽  
Nonlocal Potentials

The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.

Spiral motion for components of a Dirac system of limit circle type

1984 ◽  
Vol 97 ◽  
pp. 97-104
Author(s):  
S. G. Halvorsen ◽  
J. K. Shaw
Keyword(s):  
Complex Plane ◽  
Principal Result ◽  
Dirac System ◽  
Limit Circle ◽  
Spiral Motion ◽  
Dirac Systems ◽  
Continuous Complex ◽  
Sturm Liouville

SynopsisThe components of a two (complex) dimensional Dirac system are studied as trajectories in the complex plane. The system, defined on an interval [a, b) of regular points, is assumed to be of limit circle type at t = b, and to be non-oscillatory there. By introducing moduli ρ1(t) = |y1(t)|, ρ2(t) = |y2(t)| and continuous complex arguments θ1(t) = arg y1(t), θ2(t) = arg y2(t) for the components, the principal result proved is that θ1(t) and θ2(t) are bounded as t → b. Examples show that monotonicity of the argument function θ1(t), which is a feature of the corresponding problem for Sturm–Liouville equations, fails for Dirac systems.

scholarly journals Computation of Spectral Parameter of Discontinuous Dirac Systems with a Gaussian Multiplier

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Mohammed M. Tharwat ◽  
Mohammed A. Alghamdi
Keyword(s):  
Boundary Conditions ◽  
Spectral Parameter ◽  
Worked Examples ◽  
New Technique ◽  
Dirac System ◽  
Sinc Method ◽  
Dirac Systems

We aim in this paper to apply a sinc-Gaussian technique to compute the eigenvalues of a Dirac system which has a discontinuity at one point and contains a spectral parameter in all boundary conditions. We establish the needed properties of eigenvalues of our problem. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.

Titchmarsh–Weyl theory for q-Dirac systems

2019 ◽  
Vol 22 (02) ◽  
pp. 1950010
Author(s):  
Bilender P. Allahverdiev ◽  
Hüseyin Tuna
Keyword(s):  
Dirac System ◽  
Limit Circle ◽  
Dirac Systems ◽  
Weyl Theory ◽  
Limit Circle Case ◽  
Circle Case

In this work, we establish Titchmarsh–Weyl theory for singular [Formula: see text]-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to [Formula: see text]-analogue of this system. We show that it does not occur for the limit-circle case for the [Formula: see text]-Dirac system.

Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

2018 ◽  
Vol 18 (1) ◽  
pp. 87-104
Author(s):  
Xu Yang
Keyword(s):  
Dirac Operator ◽  
Growth Rates ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Spin Manifold ◽  
Dirac System ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Dirac Systems ◽  
Spin Manifolds

AbstractIn this article, we study the existence of solutions for the Dirac system\left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right.whereMis anm-dimensional compact Riemannian spin manifold,{u,v\in C^{\infty}(M,\Sigma M)}are spinors,Dis the Dirac operator onM, and the fiber preserving map{H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}}is a real-valued superquadratic function of class{C^{1}}with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.

scholarly journals Discrete spectrum of perturbed Dirac systems with real and periodic coefficients

1990 ◽  
Vol 115 (3-4) ◽  
pp. 337-347
Author(s):  
Boris Buffoni
Keyword(s):  
Discrete Spectrum ◽  
Upper Bound ◽  
Dirac System ◽  
Periodic Coefficients ◽  
Dirac Systems

SynopsisThis paper deals with the number of eigenvalues which appear in the gaps of the spectrum of a Dirac system with real and periodic coefficients when the coefficients are perturbed. The main results provide an upper bound and a condition under which exactly one eigenvalue appears in a given gap.

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