Relativistic stability of matter. I (Q1081912)

The quantum-mechanical system consisting of M nuclei and N electrons interacting by Coulomb forces is studied, in which special relativity is taken into account by modifying the kinetic energy of the system, so that the Hamiltonian of the system becomes \[ H=\sum^{N}_{k=1}(- \Delta_{x_ k})^{1/2}+\alpha [\sum_{j<k}\frac{1}{| x_ j-x_ k|}+\sum_{j<k}\frac{Z_ jZ_ k}{| y_ j-y_ k|}- \sum_{j,k}\frac{Z_ k}{| x_ j-y_ k|}] \] where \(Z_ i\) denote the charges of M nuclei fixed at points \(y_ i\in {\mathbb{R}}^ 3\) \((i=1,2,...,M)\). The spin is not taken into account. Then the result is obtained that a system governed by the Hamiltonian H above can implode for some unfavorable values of \(Z_ i\) and of the fine structure constant \(\alpha\), and the values of \(Z_ i\) and \(\alpha\) for which the system is stable are found. The main results of this paper are as follows. (1) If all \(Z_ i=1\), then the Hamiltonian H above is stable for \(\alpha\leq 1/2.06\pi\). (2) There exists \(\alpha_{critical}\) such that \(\alpha <\alpha_{critical}\) and \(\alpha\max_{k} Z_ k\leq 2/\pi\) imply stability of H.