dorsal/arxiv
View SchemaFurther generalization and numerical implementation of pseudo-time Schroedinger equations for quantum scattering calculations
| Authors | V. A. Mandelshtam, A. Neumaier |
|---|---|
| Categories | |
| ArXiv ID | physics/0204049 |
| URL | https://arxiv.org/abs/physics/0204049 |
Abstract
We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme. The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schroedinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U. Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications. We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states.
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"abstract": "We review and further develop the recently introduced numerical approach for\nscattering calculations based on a so called pseudo-time Schroedinger equation,\nwhich is in turn a modification of the damped Chebyshev polynomial expansion\nscheme.\n The method utilizes a special energy-dependent form for the absorbing\npotential in the time-independent Schroedinger equation, in which the complex\nenergy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the\neigenvalues of some explicitly known sparse matrix U.\n Most importantly for the numerical implementation, all the physical\neigenvalues u_k are extreme eigenvalues of U, which allows one to extract these\neigenvalues very efficiently by harmonic inversion of a pseudo-time\nautocorrelation function using the filter diagonalization method. The\ncomputation of 2T steps of the autocorrelation function requires only T sparse\nreal matrix-vector multiplications.\n We describe and compare different schemes, effectively corresponding to\ndifferent choices of the energy-dependent absorbing potential, and test them\nnumerically by calculating resonances of the HCO molecule. Our numerical tests\nsuggest an optimal scheme that provide accurate estimates for most resonance\nstates.",
"arxiv_id": "physics/0204049",
"authors": [
"V. A. Mandelshtam",
"A. Neumaier"
],
"categories": [
"physics.chem-ph"
],
"title": "Further generalization and numerical implementation of pseudo-time Schroedinger equations for quantum scattering calculations",
"url": "https://arxiv.org/abs/physics/0204049"
},
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