The local fractional Schr？dinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative. 1. Introduction As it is known, in classical mechanics, the equations of motions are described as Newton’s second law, and the equivalent formulations become the Euler-Lagrange equations and Hamilton’s equations. In quantum mechanics, Schr？dinger's equation for a dynamic system like Newton's law plays an important role in Newton's mechanics and conservation of energy. Mathematically, it is a partial differential equation, which is applied to describe how the quantum state of a physical system changes in time [1, 2]. In this work, the solutions of Schr？dinger equations were investigated within the various methods [3–12] and other references therein. Recently, the fractional calculus [13–30], which is different from the classical calculus, is now applied to practical techniques in many branches of applied sciences and engineering. Fractional Schr？dinger's equation was proposed by Laskin [31] via the space fractional quantum mechanics, which is based on the Feynman path integrals, and some properties of fractional Schr？dinger's equation are investigated by Naber [32]. In present works, the solutions of fractional Schr？dinger equations were considered in [33–38]. Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus) [39–56] is best candidate and has been applied to model the practical problems in engineering, which are nondifferentiable functions. For example, the systems of Navier-Stokes equations on Cantor sets with local fractional derivative were discussed in [42]. The local fractional Fokker-Planck equation was investigated in [43]. The basic theory of elastic problems was considered in [44]. The anomalous diffusion with local fractional derivative was researched in [48–50]. Newtonian mechanics with local fractional derivative was proposed in [51]. The fractal heat transfer in silk cocoon hierarchy and heat conduction in a semi-infinite fractal bar were presented in [53–55] and other references therein. More recently, the local fractional Schr？dinger equation in three-dimensional Cantorian system was considered in [56] as where the local fractional Laplace operator is [39, 40, 42] the
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