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  • Open Access

Analytic solution for a class of multiscale long-range potentials

Haixiang Fu1,*, Xiaoji Zhou2, and Mingzhe Li3,†

  • 1College of Information Science and Engineering, Huaqiao University, Xiamen, Fujian 361021, China
  • 2State Key Laboratory of Photonics and Communications, School of Electronics, Peking University, Beijing 100871, China
  • 3Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen, Fujian 361005, China

  • *Contact author: hxfu@hqu.edu.cn
  • †Contact author: mzli@xmu.edu.cn
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Phys. Rev. Research 7, 043096 – Published 23 October, 2025

DOI: https://doi.org/10.1103/b8sh-1dk4

Abstract

We present a unified exact analytic solution of the radial Schrödinger equation with a class of four long-range potentials, three of which are two-scale. The physics of different potentials are encapsulated in the solution variable and parameters. Based on this solution and its asymptotic behaviors, we generalize Bo Gao’s formulation of quantum defect theory for single-scale potentials to two-scale regime. As an application of this two-scale quantum defect theory, we propose a concept of harmonic oscillator quantum defect of energy levels in a spherical harmonic trap, which characterizes the effect of the two-body interactions on their trap states. This concept is a generalization of the standard quantum defect for Coulomb potential, with a distinct feature that the equations followed by the two types of quantum defects induced by the potentials under study are formally the same yet with different parameters. Examples of these two quantum defects are illustrated for Rb and Cs atoms.

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Phys. Rev. Research 7, 043096– Published 23 October, 2025
  • Received 20 July 2025
  • Accepted 17 September 2025
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