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Application of regularization maps to quantum mechanical systems in two and three dimensions

E. Harikumar

School of Physics, University of Hyderabad, Central University, P. O. Hyderabad 500046, Telangana, India

E-mail Address: eharikumar@uohyd.ac.in

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Suman Kumar Panja

School of Physics, University of Hyderabad, Central University, P. O. Hyderabad 500046, Telangana, India

E-mail Address: sumanpanja19@gmail.com

Corresponding author.

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Partha Guha

Department of Mathematics, Khalifa University of Science and Technology, P. O. Box 127788, Abu Dhabi, UAE

E-mail Address: partha.guha@ku.ac.ae

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https://doi.org/10.1142/S0217732322500432Cited by:0 (Source: Crossref)

Abstract

In this paper, we generalize the application of the Levi-Civita (L-C) and Kustaanheimo–Stiefel (K-S) regularization methods to quantum mechanical systems in two and three dimensions. Schrödinger equations in two and three dimensions, describing a particle moving under the combined influence of $\frac{1}{r}$ $\frac{1}{r}$ and $r^{2}$ $r^{2}$ potentials are mapped to that of a harmonic oscillator with inverted sextic potential, and interactions, in two and four dimensions, respectively. Using the perturbative solutions of the latter systems, we derive the eigen spectrum of the former systems. Using Bohlin–Sundmann transformation, a mapping between the Schrödinger equations describing shifted harmonic oscillator and H-atom is also derived. Exploiting this equivalence, the solution to the former is obtained from the solution of the latter.

Keywords:

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