Application of B-splines to determining eigen-spectrum of Feshbach molecules

The B-spline basis set method is applied to determining the rovibrational eigen-spectrum of diatomic molecules. A particular attention is paid to a challenging numerical task of an accurate and efficient description of the vibrational levels near the dissociation limit (halo-state and Feshbach molecules). Advantages of using B-splines are highlighted by comparing the performance of the method with that of the commonly-used discrete variable representation (DVR) approach. Several model cases, including the Morse potential and realistic potentials with 1/R^3 and 1/R^6 long-range dependence of the internuclear separation are studied. We find that the B-spline method is superior to the DVR approach and it is robust enough to properly describe the Feshbach molecules. The developed numerical method is applied to studying the universal relation of the energy of the last bound state to the scattering length. We numerically illustrate the validity of the quantum-defect-theoretic formulation of such a relation for a 1/R^6 potential.

💡 Research Summary

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The paper presents a comprehensive study of using B‑spline basis functions to calculate the rovibrational eigen‑spectrum of diatomic molecules, with a particular focus on the challenging regime near the dissociation limit where halo‑type and Feshbach molecular states appear. The authors first motivate the use of B‑splines by highlighting their completeness, locality, and the ability to achieve high accuracy with relatively few basis functions compared with traditional finite‑difference or discrete variable representation (DVR) methods.

Methodologically, the radial Schrödinger equation for nuclear motion is treated via a Galerkin approach. The wavefunction is expanded in a set of B‑splines of order k (typically k = 15) defined on a non‑uniform knot sequence. To enforce the physical boundary conditions u(R_min)=u(R_max)=0, the first and last splines are omitted, leaving n‑2 active coefficients. This expansion leads to a generalized eigenvalue problem A c = E B c, where A contains kinetic‑energy and potential contributions and B is the overlap matrix. Because B‑splines have compact support, both matrices are sparse, allowing efficient diagonalization with Lanczos or Arnoldi algorithms.

A crucial ingredient is the “mapped grid” technique, which rescales the radial coordinate according to the local de Broglie wavelength. The mapping function
x(R) = β⁻¹ √(2μ) ∫_{R_min}^{R}√