Extension of perturbation theory to quantum systems with conformable derivative

In this paper, the perturbation theory is extended to be applicable for systems containing conformable derivative of fractional order ?. This is needed as an essential and powerful approximation method for describing systems with conformable differential equations that are difficult to solve analytically. The work here is derived and discussed for the conformable Hamiltonian systems that appears in the conformable quantum mechanics. The required ?-corrections for the energy eigenvalues and eigenfunctions are derived. To demonstrate this extension, three illustrative examples are given, and the standard values obtained by the traditional theory are recovered when ?=1.