One‐dimensional q‐Dirac equation

Abstract

In this paper, we introduce a q-analog of 1-dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of L 2 q ((0, a) ; E). Finally, we give some examples. KEYWORDS eigenfunction expansions, eigenvalues and eigenfunctions, Green matrix, q−Dirac operator, self-adjoint operator Math Meth Appl Sci. 2017;40:7287-7306.wileyonlinelibrary.com/journal/mma

Conformable fractional Sturm‐Liouville equation

“…Conversely, the function (t, ) defined by (32) satisfies (29) and ( 30)- ( 31). Green's function G (t, x, ) is unique in the sense that if there exists another functionG(t, x, ) such that (32) is satisfied, then…”

“…A similar way was employed earlier in the q−difference equation case in other studies. 32,33 In the study of Al-Refai and Abdeljewad, 34 the authors discussed the Sturm-Liouville problems in the frame of conformable derivatives. They prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and they establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue.…”

Inverse problems for one dimensional conformable fractional Dirac type integro differential system

“…A variety of new results can be found ( [2], [3] and references therein). In recent years, scholars have focussed on a fractional generalization of the well known Sturm-Liouville and Dirac problems ( [4], [5], [6], [7], [8], [16], [27], [28] and [36]).…”

Spectral properties and a Parseval’s equality in the singular case for q-Dirac problem

“…[10,14,16,22,[25][26][27]30]. In [2,3], a q-analog of one dimensional Dirac system on a finite interval was investigated and the authors studied the existence and uniqueness of its solution, and some spectral properties. Also, the asymptotic formulas for the eigenvalues and the eigenfunctions were obtained in [18].…”

“…x)f η,i (x)2 d q x = 0, i = 1, 2.Then ∞ 0 f 2 η,1 (x) + f 2 η,2 (x) d q x = ∞ -∞ F 2 η,1 (λ) + F 2 η,2 (λ) dρ(λ),whereF η,i (λ) = ∞ 0 f η,i (x)Y i (x, λ) d q x. Since for i = 1, 2, ∞ 0 f η 1 ,i (x)f η 2 ,i (x) 2 d q x → 0 as η 1 , η 2 → ∞, we get ∞ -∞ F η 1 ,i (λ) -F η 2 ,i (λ) 2 dρ(λ) = ∞ 0 f η 1 ,i (x)f η 2 ,i (x) 2 d q x → 0, i = 1, 2,as η 1 , η 2 → ∞.…”