Abstract
In this paper we consider the problem of the occurrence of spurious modes when computing the eigenvalues of Dirac operators, with the motivation to describe relativistic electrons in an atom or a molecule. We present recent mathematical results which we illustrate by simple numerical experiments. We also discuss open problems.
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Notes
- 1.
In our examples we will have \(\mathfrak {H}=L^2(\Omega )\), the space of square-integrable functions on a domain \(\Omega \) in the \(N\)-dimensional space \({\mathbb R }^N\). We will encounter two main cases: that of the whole physical space \(\Omega ={\mathbb R }^3\) and that of the half line \(\Omega =(0,\infty )\) useful to deal with radial functions.
- 2.
Take for instance \(\psi _n(\mathbf {r})=\exp (i\mathbf {p}\cdot \mathbf {r}/\hbar )n^{-N/2}\chi (\mathbf {r}/n)\) for some smooth \(\chi \) with \(\int _{{\mathbb R }^N}|\chi (\mathbf {r})|^2\mathrm{d}^Nr=1\) and a momentum \(\mathbf {p}\) such that \(p^2=2m\lambda \).
- 3.
Sometimes the basis is rather taken to be \(\sigma _k\partial _k\varphi _n\), which multiplies the number of lower spinors by 3.
- 4.
Actually, in [20], the function is taken of the form \(\varphi _n=\left( f(nr^2)+g(\delta n r^2)\right) {1\atopwithdelims ()0}\) where \(f\) and \(g\) are chosen with disjoint support, which simplifies some calculations.
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Acknowledgments
M. L. would like to thank Lyonell Boulton and Nabile Boussaid for stimulating discussions, in particular concerning the numerical experiments of this article. M. L. has received financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement MNIQS 258023). M. L. and É. S. acknowledge financial support from the French Ministry of Research (ANR-10-BLAN-0101).
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Lewin, M., Séré, É. (2014). Spurious Modes in Dirac Calculations and How to Avoid Them. In: Bach, V., Delle Site, L. (eds) Many-Electron Approaches in Physics, Chemistry and Mathematics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-06379-9_2
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