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The local fractional Schrödinger 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.

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, Schrödinger'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 [

Recently, the fractional calculus [

Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus) [

More recently, the local fractional Schrödinger equation in three-dimensional Cantorian system was considered in [

with

The local fractional Schrödinger equation in two-dimensional Cantorian system can be written as

With the potential energy

In this paper our aim is to investigate the nondifferentiable solutions for local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method [

According to local fractional series expansion method [

In view of (

There are nondifferentiable terms, which are written as

In view of (

Therefore,

Then, following (

Let

So, we have

In [

Here, we consider the following operator:

Using the iterative formula (

Let us change (

With

so that

Using iteration relation (

and an initial value is given by

Therefore, following (

and so on.

Hence, from (

We transform (

The initial condition is presented as

Applying (

and so forth.

Hence, we have the nondifferentiable solution of (

In the work, we have obtained the nondifferentiable solutions for the local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method. The present method is shown that is an effective method to obtain the local fractional series solutions for the partial differential equations within local fractional differentiable operator.

This work was supported by Natural Science Foundation of Hebei Province (no. F2010001322).