Applications of Discrete Singular Convolution Algorithm in One-Electron Systems: Hydrogen Atom

Abstract

The application of discrete singular convolution (DSC) algorithm in solving the Schrödinger equation of one-electron systems was attempted with a hydrogen atom as an example. Using the uniform discretization and Shannon kernel, the Schrödinger equations were solved in spherical coordinates. Compared with other methods such as discrete variable representation, Hartree-Fock, and density functional theory, the DSC algorithm is robust and efficient in numerical solutions for achieving accurate eigenvalues of excited states, and thus, especially suitable for problems in which lots of accurate eigenvalues of excited states are requested. The restrictions of boundary conditions on the discretization and the impact of singularities are studied. Both the boundary conditions and singularities were found to be critical in the numerical applications.

Comparison of Delta-Type Discrete Singular Convolution Kernels for Anti-noise Edge Detection

“…DSC was developed as a powerful tool for singular integration [1], wavelet generation [2], vibration analysis [3], solving different equations [4][5][6], nonlinear transient analysis [7] and so forth. In the context of image processing, DSC is used to facilitate anti-noise edge detector based on Shannon's delta kernel which provides a relatively efficient way for image noise filtering so as to obtain correct edges [8].…”

A generalized discrete singular convolution algorithm improved by regularizing singularities for one electron system

“…should have no effect on the interval which is kept. Physically, the potentials on the abandoned intervals are set to be infinity and therefore the values of the wavefunction are forced to be zero [26,27].…”

“…In recent years, a discrete singular convolution (DSC) algorithm has been proposed [19] and applied in various engineering systems without singularities and obtained competitive results [20][21][22][23][24][25]. Therefore we have been encouraged to apply the DSC in one-electron physical systems [26,27]. The DSC has been used to solve the radial Schrödinger equation of a hydrogen atom [26].…”

“…Therefore we have been encouraged to apply the DSC in one-electron physical systems [26,27]. The DSC has been used to solve the radial Schrödinger equation of a hydrogen atom [26]. However, since the results in [26] are affected distinctly by the Coulomb singularity, we introduced a mapping method to obtain a nonuniform discretization to regularize the singularity [27].…”

“…The DSC has been used to solve the radial Schrödinger equation of a hydrogen atom [26]. However, since the results in [26] are affected distinctly by the Coulomb singularity, we introduced a mapping method to obtain a nonuniform discretization to regularize the singularity [27]. The mapping procedure is used to produce a nonuniform discretization which is appropriate to the singular potential from a uniform discretization, but with the form of the potential and Hamiltonian unchanged.…”