The quantum-mechanical Coulomb propagator in an L2 function representation
Abstract The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Nature Portfolio
2021-09-01
|
| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-021-96925-0 |
| _version_ | 1818858423648780288 |
|---|---|
| author | Rolf Gersbacher John T. Broad |
| author_facet | Rolf Gersbacher John T. Broad |
| author_sort | Rolf Gersbacher |
| collection | DOAJ |
| description | Abstract The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L $$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures. |
| first_indexed | 2024-12-19T08:56:03Z |
| format | Article |
| id | doaj.art-c219527a745f4e418a373e11a2abe5f3 |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-12-19T08:56:03Z |
| publishDate | 2021-09-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-c219527a745f4e418a373e11a2abe5f32022-12-21T20:28:37ZengNature PortfolioScientific Reports2045-23222021-09-0111111610.1038/s41598-021-96925-0The quantum-mechanical Coulomb propagator in an L2 function representationRolf Gersbacher0John T. Broad1University of Applied Sciences EsslingenUniversity of Applied Sciences EsslingenAbstract The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L $$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.https://doi.org/10.1038/s41598-021-96925-0 |
| spellingShingle | Rolf Gersbacher John T. Broad The quantum-mechanical Coulomb propagator in an L2 function representation Scientific Reports |
| title | The quantum-mechanical Coulomb propagator in an L2 function representation |
| title_full | The quantum-mechanical Coulomb propagator in an L2 function representation |
| title_fullStr | The quantum-mechanical Coulomb propagator in an L2 function representation |
| title_full_unstemmed | The quantum-mechanical Coulomb propagator in an L2 function representation |
| title_short | The quantum-mechanical Coulomb propagator in an L2 function representation |
| title_sort | quantum mechanical coulomb propagator in an l2 function representation |
| url | https://doi.org/10.1038/s41598-021-96925-0 |
| work_keys_str_mv | AT rolfgersbacher thequantummechanicalcoulombpropagatorinanl2functionrepresentation AT johntbroad thequantummechanicalcoulombpropagatorinanl2functionrepresentation AT rolfgersbacher quantummechanicalcoulombpropagatorinanl2functionrepresentation AT johntbroad quantummechanicalcoulombpropagatorinanl2functionrepresentation |