Coordinate bending studies for univariate Schr\"odinger equation: Cubic and inverse cubic bending functions

Abstract

This paper is designed to present a comparative analysis of cubic  and inverse cubic coordinate bending. We have shown that the cubic  case of bending to get a post bending Hamiltonian whose potential
is just harmonic with an arbitrary harmonicity provides a characteristic potential of infinite well type as desired. However, the absence of sufficient parametrization and the full matrix structure of the weight over Hermite functions causes unwillingness for further progress. On the other hand, the inverse cubic coordinate bending case does not give infinite well structure in the characteristic potential as long as only harmonic oscillator structure in the post bending Hamiltonian is insisted on. To escape from this reality we add polynomial anharmonicity with arbitrary but positive coefficients to the post bending Hamiltonian's potential part. This results in finite multidiagonality for main and weight operators when the scaled Hermite function based matrix representative  algebraic eigenvalue equation is obtained. The polynomial anharmonicities can be used to enforce the characteristic potential to approximate the true given potential. The harmonicity coefficient
(elastic force constant) can be used as an asymptotic perturbation agent to develope perturbative approximant at infinite harmonicity (where elastic force constant goes to infinity). We conclude that inverse cubic coordinate bending can serve us to get approximate solutions of Schr\"odinger equation for obtaining system energies and for solving similar problems. This may also be extended to some other coordinate bendings.