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Certain Perturbative Spectral Investigations For An Integrated Symmetric Exponential Well Potential Quantum System Via Coordinate Bending From Quantum Harmonic Oscillator System

Last modified: 2020-01-28

#### Abstract

For most of the quantum systems determining the spectral properties is not easy. Even as a result of intense efforts, it is often not possible to get exact solution. The use of approximation methods comes to the fore here. Perturbation theory is one of the straightforward and handy methods in these situations. Meaningful and interpretable results can be obtained by using the solutions of known systems. However, in order to apply this method, first convenient equations must be obtained. There are several studies performed in our working group to develop methodologies for these purpose. This work aims to develop a new method based on a coordinate bending on the target system whose potential is an harmonic oscillator with an arbitrary elastic force constant. Coordinate bending starts from this system's Schrödinger equation and a factor from the wave function is extracted in such a way that the resulting new ODE contains basically a self adjoint Hamiltonian. The potential function in our target system is chosen via an appropriate coordinate bending function. We desire to focus on such a potential that it increases with a symmetric exponential containing function structure. We prefer to use not the exponential function but its integral. The parametrization is also realized in such a way that the auxiliary ODE eigenfunctions do not contain the arbitrary elastic force constant. In the solution procedure we use a perturbation expansion and we give the zeroth and first order component determinations and also we present a somehow brief instruction to proceed to higher order perturbation components. The arbitrary elastic force constant is determined to suppress the norm of the perturbation operator with respect to auxiliary operator's eigenfunction. The minimum norm after this determination appears to be the fluctuation in the target Hamiltonian.