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Lectures on large Coulomb systems. (English) Zbl 0830.35111
Feldman, J. (ed.) et al., Mathematical quantum theory II: Schrödinger operators. Proceedings of the Canadian Mathematical Society annual seminar on mathematical quantum theory held in Vancouver, Canada, August 4-14, 1993. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 8, 73-107 (1995).
The author studies here the ground state energy of a molecule consisting of \(N\) electrons and \(M\) nuclei of charges \(Z \equiv (Z_1, \ldots, Z_M)\) at \(R \equiv (R_1, \ldots, R_M)\). The Schrödinger operator of such a system is \[ H(Z,R) = \sum^N_{i = 1} \bigl( - \Delta_i/2 - V(x_i, Z,R) \bigr) + \sum_{i < j} |x_i - x_j |^{-1}; \;V(x,Z,R) = \sum^M_{i = 1} Z_i/ |x - R_i |. \] He obtains \[ E(Z,R) \equiv \inf \text{spec} H(Z,R) = - a_{TF} |Z |^{7/3} + \sum^M_{i = 1} Z_i^2/2 + O \bigl( a^{-1} |Z |^{16/9 + \delta} \bigr),\;\delta > 0, \] as \(Z \to \infty\) in \(\mathbb{R}^M\). Here \(a_{TF} \geq 0\) is the constant derived from Thomas-Fermi energy, \(a = \min (|Z |^{1/3} |R_i - R_j |,\;i \neq j ;1)\), and \(|Z |= \sum^M_{i = 1} Z_i = N\). The author investigates the ground state energy \(E_{\text{ind}} (Z,R)\) of the mean field Hamiltonian with Thomas-Fermi potential, and proves his results from the estimate of \(E(Z,R) - E_{\text{ind}} (Z,R)\). Next he obtains \[ IE \equiv (\text{inf contrspec}) H - (\text{inf spec}) H < C |Z |^{20/21} \] for ionization energy \(IE\) etc. Finally he gives various open problems relating to his theme.
For the entire collection see [Zbl 0815.00011].
Reviewer: H.Yamagata (Osaka)

35Q40 PDEs in connection with quantum mechanics
35P15 Estimates of eigenvalues in context of PDEs
81V55 Molecular physics
35J10 Schrödinger operator, Schrödinger equation
81V70 Many-body theory; quantum Hall effect
81Q15 Perturbation theories for operators and differential equations in quantum theory