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Zbl 1062.35100
Yserentant, Harry
On the regularity of the electronic SchrÃ¶dinger equation in Hilbert spaces of mixed derivatives.
(English)
[J] Numer. Math. 98, No. 4, 731-759 (2004). ISSN 0029-599X; ISSN 0945-3245/e

The author studies the eigenfunctions $u(x)= \psi(x,\sigma)$; $x\in \Bbb R^3)^N$, $\sigma\in \{-1/2,1/2\}^N$ (spin) of the Hamilton operator $$H=- \sum_{i=1\sim N}\Delta_i/2- \sum_{i=1\sim N}\sum_{\nu= 1\sim K}Z_\nu/|x_i- a_\nu|+ \sum_{i,j= 1\sim N,i\ne j}(1/2)/|x_i- x_j|,$$ under the condition $$\psi(Px, P\sigma)= \text{sign}(P)\psi(x, \sigma),\quad P\sigma= \sigma.$$ Let $\emptyset\ne I\subseteq \{1,2,\dots, N\}$, $P$ an exchange of two indices in $I$, and $D_I(\subset D((\Bbb R^3)^N))$ be the space of antisymmetric functions under $\forall P$ in $I$. Let $I^*$ be the set of the mappings $\alpha: I\to \{1,2,3\}$, $L_\alpha= \prod_{i\in I}\partial/\partial x_{i,\alpha(i)}$, $\alpha\in I^*$, and $$\Vert u\Vert_{I,s}= \Biggl(\Vert u\Vert^2_s+ \sum_{\alpha\in I^*}\Vert L_\alpha u\Vert^2_s\Biggr)^{1/2},$$ $s=-1$, $0$, or $1$. The completion of $D_I$ under $\Vert\cdot\Vert_{I,s}$ (in $H^s$) is denoted as $X_I^s$ $(H_I^s)$.\par Result: (1) Let $\chi= v+(-1)^{|I|}\prod_{i\in I}\Delta_i v$. If $\mu> \mu_0(N)$, the solution $u\in H^1_I$ of the equation $((H+ \mu I) u,\chi)= (f,\chi)$, $\chi\in H^1_I$ is contained in $X^1_I$ for all $f\in X^{-1}_I$, $\Vert u\Vert_{I,1}\le 4\Vert f\Vert_{I,-1}$. That is, $\partial(L_\alpha u)/\partial x_{i,j}\in L^2$ ($i= 1,2,\dots, N$, $j= 1,2,3$) for $\forall\alpha\in I^*$ (regularity). (2) Let $(P_k u)(x)= (2\pi)^{-3N/2} \int \chi_k(\omega)(Fu)(\omega)\exp(i\omega\cdot x)\,d\omega$, $\chi_k(\omega)= 1$, if $\prod_{i= 1\sim N}|\omega_i|< 2^k$; $=0$, otherwise. The error estimate $\Vert u- P_k u\Vert_1\le 2^{-k/2}|||u|||_1$ is given.
[Hideo Yamagata (Osaka)]
MSC 2000:
*35Q40 PDE from quantum mechanics
81V10 Electromagnetic interaction
81Q10 Selfadjoint operator theory in quantum theory
35B65 Smoothness of solutions of PDE
41A63 Multidimensional approximation problems
46E20 Hilbert spaces of functions defined by smoothness properties

Keywords: Coulomb interaction; quantum chemistry; regularity; wavefunctions; derivatives; approximation methods

Cited in: Zbl 1116.78007 Zbl 1084.65125

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