12th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA), Armenia, 3 - 06 July 2018, vol.2046
When working on a Schrodinger equation, finding a general method which gives exact solution for all systems is nearly impossible. The superiorities of methods are discussed relative to each other have been examined in numerous studies [1-3]. But it is easy for the systems defined under certain conditions just like one dimensional harmonic oscillator. Simulating them in the analysis will be beneficial. In this work, we are trying to solve the univariate Schrodinger equation for our specially-defined system, exponentially anharmonic symmetric oscillator, by using approximation methods. Analyses show that, position variable, x -> infinity limit gives clues for the asymptotic behavior of the solution of the spectral equation of Hamilton operator. But in our work, we prefer to use a new proper function as a new coordinate instead of x to facilitate the analysis. This function defines actually a coordinate under certain conditions. We call this function definition as "Coordinate Axis Bending" or shortly "Coordinate Bending"[4, 5]. In the solution step, we try basis set of eigenfunctions of harmonic oscillator which is defined with the help of Hermite polynomials [6]. The recursive relation between the Hermite polynomials can simplify the computation of eigenvalue problem of our specified system. The numerical complications of overall computations lead us to use "Mathematical Fluctuation Theory" which approximates to an operator with the use of new matrix representations [7-10].