Abstract
In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green’s function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2ω,q((ω0, a); E).
Highlights
The Dirac system y10 (x) y20 (x) +p (x) 0 0 r (x) y1 (x) y2 (x) ; x 2 [a; b]; (1)where p (:) and r (:) are real-valued, Lebesgue integrable functions on [a; b], is one of the classical equations in quantum mechanics
Where p (:) and r (:) are real-valued, Lebesgue integrable functions on [a; b], is one of the classical equations in quantum mechanics. This system predicts the existence of antimatter and describes the electron spin
Some properties of the one-dimensional Dirac systems have been considered in the literature
Summary
Where p (:) and r (:) are real-valued, Lebesgue integrable functions on [a; b], is one of the classical equations in quantum mechanics. The authors study the existence and uniqueness of solutions for the initial value problems for Hahn di¤erence equations. With boundary conditions k11y1 (!0) + k12y2 (!0) = 0; k21y1 (a) + k22y2 h 1 (a) = 0; where is a complex eigenvalue parameter, kij 2 R (i; j = 1; 2) and p (:) ; r (:) are real-valued continuous functions at !0; de...ned on [!0; a]: Hira investigated the existence and uniqueness of solutions for this problem and gave its spectral properties. Hahn-Dirac boundary-value problem (2), such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green’s function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2!;q((!0; a); E). We construct the associated Green function of the Hahn-Dirac equation and give the eigenfunction expansions
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