Abstract
We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale mathbb{T}. Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on mathbb{T} . So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory.
Highlights
Fractional calculus means differentiation and integration of a noninteger order
Before expressing the conformable fractional (CF) derivative of order α ∈
We consider below the CF Dirac eigenvalue problem on an arbitrary time scale
Summary
Fractional calculus means differentiation and integration of a noninteger order. The idea of fractional calculus was introduced by Leibniz and L’Hopital in. The work of combining fractional calculus and time scale calculus in spectral theory is much less extensive To fill this gap, we consider below the CF Dirac eigenvalue problem on an arbitrary time scale. The Dirac operator is the relativistic Schrödinger operator in quantum physics It is a modern presentation of the relativistic quantum mechanics of electrons intended to make new mathematical results accessible to a wider audience. It treats in some depth the relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states, and the external field problem in quantum electrodynamics, without neglecting the interpretational difficulties and limitations of the theory.
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