Reality Of Eigenvalues Research Articles

Diamond-Type Dirac Dynamic System in Mathematical Physics

In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (⋄α) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior—which does not occur in the real line or on discrete time scales—are analyzed using this combined concept. In this study, we consider a ⋄α Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the ⋄α Dirac boundary value problem (BVP) on a uniform time scale.

Proportional fractional Dirac dynamic system

<abstract><p>In this study, considering the proportional fractional derivative, which is a generalization of the conformable fractional derivative, we provided some important spectral properties such as the reality of eigenvalues, the orthogonality of eigenfunctions, the self-adjointness of the operator, the asymptotic estimations of eigenfunctions, and Picone's identity for a proportional Dirac system on an arbitrary time scale. We also presented graphics representing the eigenfunctions of the Dirac system on a time scale, produced by taking advantage of the proportional fractional derivative with some special cases. The main purpose of presenting these graphics was to examine the effect of the proportional fractional derivative on the Dirac system on a time scale, as well as the effect of the eigenvalues, which are meaningful for the subject we were studying for the solution functions.</p></abstract>

Spectral structure and solution of fractional hydrogen atom difference equations

In this study, discrete fractional hydrogen atom (DFHA) operators are presented. Hydrogen atom differential equations have series solution due to having singularity and also obtaining a series solution for DFHA equations have some difficulties, for this reason, we study to obtain solution of DFHA equations by means of nabla Laplace transform. In addition to all these, we show self-adjointness of the DFHA operator and some spectral properties, like orthogonality of distinct eigenfunctions, reality of eigenvalues. Finally, we find an analytical solution of the problem under different q (t) potential functions, different fractional orders and different eigenvalues and the results obtained are illustrated by tables and simulations.

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.