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High-Order Matrix Numerov for Singular Potentials

Nir Barnea

Abstract

The matrix Numerov method provides an efficient framework for solving the time-independent Schrödinger equation as a matrix eigenvalue problem. However, for singular potentials such as the Coulomb interaction, the expected fourth-order convergence deteriorates for low angular momenta due to the behavior of the potential near the origin. We show that this loss of accuracy originates from an implicit boundary assumption in the standard formulation. By incorporating analytic near-origin information into the discretized Hamiltonian, we derive simple boundary corrections that restore fourth-order convergence and can even produce higher convergence rates for $s$- and $p$-wave energies. The resulting scheme preserves the simplicity and computational efficiency of the original method while significantly improving its accuracy for singular potentials.

High-Order Matrix Numerov for Singular Potentials

Abstract

The matrix Numerov method provides an efficient framework for solving the time-independent Schrödinger equation as a matrix eigenvalue problem. However, for singular potentials such as the Coulomb interaction, the expected fourth-order convergence deteriorates for low angular momenta due to the behavior of the potential near the origin. We show that this loss of accuracy originates from an implicit boundary assumption in the standard formulation. By incorporating analytic near-origin information into the discretized Hamiltonian, we derive simple boundary corrections that restore fourth-order convergence and can even produce higher convergence rates for - and -wave energies. The resulting scheme preserves the simplicity and computational efficiency of the original method while significantly improving its accuracy for singular potentials.
Paper Structure (11 sections, 42 equations, 4 figures, 2 tables)

This paper contains 11 sections, 42 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Finite Difference Method
  3. Numerov Method
  4. The singularity at the origin
  5. The $\ell = 0$ case
  6. The $\ell = 1$ Case
  7. The $\ell \ge 2$ Case
  8. Numerical Results
  9. Harmonic Oscillator
  10. The Hydrogen Atom
  11. Conclusions

Figures (4)

  • Figure 1: Convergence rate of the harmonic oscillator energy level $n=0,\ell=1$ as a function of the number of grid points. Red circles - the standard matrix Numerov method PGW12, blue triangles - Matrix Numerov with first order correction, black squares - the finite difference method. Dashed lines - fit to equation \ref{['fitq']}.
  • Figure 2: Convergence rate of the hydrogen ground state energy $n=1,\ell=0$ as a function of the number of grid points. Red circles - the original matrix Numerov method PGW12, blue triangles - Matrix Numerov with first order correction, purple squares - with second order corrections, orange triangles - with third order corrections. Dashed lines - fit to equation \ref{['fitq']}.
  • Figure 3: Convergence rate of the hydrogen energy level $n=3,\ell=0$ as a function of the number of grid points. Red circles - the original matrix Numerov method PGW12, blue triangles - Matrix Numerov with first order correction, purple squares - with second order corrections, orange triangles - with third order corrections. Dashed lines - fit to equation \ref{['fitq']}.
  • Figure 4: Convergence rate of the hydrogen energy level $n=1,\ell=1$ as a function of the number of grid points. Red circles - the original matrix Numerov method PGW12, blue triangles - Matrix Numerov with first order correction, purple squares - with second order corrections. Dashed lines - fit to equation \ref{['fitq']}.