About: Finite-difference method (FDM) is a dominant approach in the solution of differential equations by numerical means. It is ubiquitous in the solution of a large variety of problems in science and technology. Here, we present a procedure to obtain upper and lower bounds to the spectrum of the time-independent atomic-molecular-optical Hamiltonian utilizing the FDM. The procedure combines two FDM approaches with the same representation of the Laplacian. The standard FDM fixes the box size and increases the grid density with the number of grid points. In contrast, the second approach keeps the grid density constant and increases the box size with the number of points. The estimation of the error in the computation of the spectrum can be increased simply by calculating the upper and lower bounds. A number of illustrative numerical examples are given, involving the calculation of the spectra of known analytical solutions. The relevance of our finding to other grid methods is discussed.   Goto Sponge  NotDistinct  Permalink

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  • Finite-difference method (FDM) is a dominant approach in the solution of differential equations by numerical means. It is ubiquitous in the solution of a large variety of problems in science and technology. Here, we present a procedure to obtain upper and lower bounds to the spectrum of the time-independent atomic-molecular-optical Hamiltonian utilizing the FDM. The procedure combines two FDM approaches with the same representation of the Laplacian. The standard FDM fixes the box size and increases the grid density with the number of grid points. In contrast, the second approach keeps the grid density constant and increases the box size with the number of points. The estimation of the error in the computation of the spectrum can be increased simply by calculating the upper and lower bounds. A number of illustrative numerical examples are given, involving the calculation of the spectra of known analytical solutions. The relevance of our finding to other grid methods is discussed.
subject
  • Riemannian geometry
  • Fourier analysis
  • Differential geometry
  • Multivariable calculus
  • Tensors
  • Vectors (mathematics and physics)
  • Numerical differential equations
  • Differential operators
  • Finite differences
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