Determination of spectral quantities in hyperbolically Bent coordinates with characteristic potentials
This work is about coordinate bending. Coordinate bending is applied to the Schr\"odinger equation of a one-degree-of-freedom quantum system. The aim is to get a quadratic equation from the potential term of the equation. This eases the solution by imitating the Harmonic Oscillator. Before this work, several attempts have been done to do coordinate bending with polynomials but they were not adequate to get the requested function efficiently. In this paper, hyperbolic($\sinh$) function is applied as a coordinate bending function since it includes exponential forms. After getting the special form of the potential, we can arrive at a weighted eigenvalue problem. Beside the potential, this weight is determined in this study.
Then, an approach to solve the obtained eigenvalue problem has been developed and its positive/negative aspects are discussed at the end.