Determination of spectral quantities in hyperbolically Bent coordinates with characteristic potentials

Abstract

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.