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| ericmarkusnotes:start [2017/04/13 15:54] – [Kohn-Sham equations] estachura | ericmarkusnotes:start [2017/04/13 19:10] (current) – [Kohn-Sham equations] estachura |
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| ===Jerome's Paper=== | ===Jerome's Paper=== |
| * With regard to [[http://sites.math.northwestern.edu/~jwj/preprints/JMAA/JMAA.pdf|Jerome's paper]]: Can this be extended to unbounded regions? Also, why are homogeneous Dirichlet boundary values imposed? Are these the most natural? Can others be used, in particular, nonzero? Additionally, he mentions at the end a special case: dimension 1, where one might need the Hilbert transform to deal with the Hartree potential. | * With regard to [[http://sites.math.northwestern.edu/~jwj/preprints/JMAA/JMAA.pdf|Jerome's paper]]: Can this be extended to unbounded regions? Also, why are homogeneous Dirichlet boundary values imposed? Are these the most natural? Can others be used, in particular, nonzero? Additionally, he mentions at the end a special case: dimension 1, where one might need the Hilbert transform to deal with the Hartree potential. |
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| * There is also the previous result of Jerome here [[https://arxiv.org/pdf/1309.3587.pdf|Jerome's previous paper]] which is more numerical in nature. Here bounded domains with homogeneous B.C. are also studied. | * There is also the previous result of Jerome here [[https://arxiv.org/pdf/1309.3587.pdf|Jerome's previous paper]] which is more numerical in nature. Here bounded domains with homogeneous B.C. are also studied. |
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| | * 1D Hartree potential for the electron charge density $\rho = |\psi|^2$: $$V_H(x,t) = 1/|x| \ast \rho (x,t) = \int_a^b \dfrac{\rho(y,t)}{|x-y|} dy$$ and Hilbert transform for $f \in L^2 (\mathbb{R})$ is $$H f (y) = \lim_{\epsilon \to 0} \dfrac{1}{\pi} \int_{|x| > \epsilon} \dfrac{f(y-x)}{x} dx$$ Recall that $H: L^2 (\mathbb{R}) \to L^2 (\mathbb{R})$ is an isometry and for $1<p < \infty$, $H: W^{s,p}(\mathbb{R}) \to W^{s,p}(\mathbb{R})$ is an isomorphism. In the 1D case Jerome suggests considering the softened potential given (e.g. for Helium atom) by $$V(x_1, x_2) = \dfrac{1}{\sqrt{(x_1-x_2)^2+1}}$$ subject to a softened external potential $$V_{\text{ext}}(x) = \dfrac{-2}{\sqrt{x^2+1}}$$ Jerome suggests Hilbert transform is needed for this, but maybe if another type of softened potential is considered, e.g. Eric's $V(x) = \dfrac{e^{-C/|x|}}{|x|}$ no Hilbert transform is needed? |
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| | * Ionic potentials not included in Jerome's analysis, can we include them? What general form do they take? |
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| ===Sprengel Papers=== | ===Sprengel Papers=== |
