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Analyticity of the density of electronic wavefunctions. (English) Zbl 1068.35118

We consider an \(N\)-electron molecule with \(L\) fixed nuclei. The non-relativistic Hamiltonian of the molecule is given by \[ H=-\Delta+V(x), \] where \[ \Delta=\sum_{j=1}^N\Delta_j, \]
\[ V(x)=\sum_{j=1}^N\left(-\sum_{l=1}^L\frac{Z_l}{| x_j-R_l| }\right)+\sum_{1\leq i<j\leq N}\frac1{| x_i-x_j| }+\sum_{1\leq l<k\leq L}\frac{Z_lZ_k}{| R_l-R_k| }, \] is Coulomb potential, \(\Delta_j\) is \(3\)-dimensional Laplacian, \((R_1,\dots,R_L)\in \mathbb R^{3L}\), \(R_l\neq R_k\) for \(k\neq l\), denote the position of the \(L\) nuclei whose positive charges are given by \((Z_1,\dots,Z_L)\). Positions of the \(N\) electrons are denoted by \((x_1,\dots,x_N)\in \mathbb R^{3N}\), where \(x_j\) denotes the position of the \(j\)th electron in \(\mathbb R^3\). The following theorem is the main result of this article.
Let \(\psi\in L^2(\mathbb R^{3N})\) satisfy the equation \(H\psi=E\psi\), with \(E\in \mathbb R\). Then \[ \varrho(x)=\int_{\mathbb R^{3N-3}}| \psi(x,x_2\dots,x_N)| ^2\,dx_2\dots dx_N \] is a real analytic function in \(\mathbb R^3\setminus\{R_1,\dots,R_L\}\).

MSC:

35Q40 PDEs in connection with quantum mechanics
35B25 Singular perturbations in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
47N50 Applications of operator theory in the physical sciences
81V55 Molecular physics
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References:

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