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On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. (English) Zbl 1062.35100

The author studies the eigenfunctions \(u(x)= \psi(x,\sigma)\); \(x\in \mathbb R^3)^N\), \(\sigma\in \{-1/2,1/2\}^N\) (spin) of the Hamilton operator \[ H=- \sum_{i=1\sim N}\Delta_i/2- \sum_{i=1\sim N}\sum_{\nu= 1\sim K}Z_\nu/|x_i- a_\nu|+ \sum_{i,j= 1\sim N,i\neq j}(1/2)/|x_i- x_j|, \] under the condition \[ \psi(Px, P\sigma)= \text{sign}(P)\psi(x, \sigma),\quad P\sigma= \sigma. \] Let \(\emptyset\neq I\subseteq \{1,2,\dots, N\}\), \(P\) an exchange of two indices in \(I\), and \(D_I(\subset D((\mathbb R^3)^N))\) 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 \[ \| u\|_{I,s}= \Biggl(\| u\|^2_s+ \sum_{\alpha\in I^*}\| L_\alpha u\|^2_s\Biggr)^{1/2}, \] \(s=-1\), \(0\), or \(1\). The completion of \(D_I\) under \(\|\cdot\|_{I,s}\) (in \(H^s\)) is denoted as \(X_I^s\) \((H_I^s)\).
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 \(f\in X^{-1}_I\), \(\| u\|_{I,1}\leq 4\| f\|_{I,-1}\). 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 \(\| u- P_k u\|_1\leq 2^{-k/2}|||u|||_1\) is given.

MSC:

35Q40 PDEs in connection with quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35B65 Smoothness and regularity of solutions to PDEs
41A63 Multidimensional problems
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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