On the existence of solutions of the helium wave equation

Abstract

Introduction. The helium wave equation has been studied by several authors(') with a view to obtaining the exact solution for the ground state, or at least to establishing its existence. The subject is of importance as a matter of principle, since it implies the decision whether the present formulation of nonrelativistic many-body problems is correct(2). Also, the validity of the variational method(3) depends upon it, for this method would be meaningless if there existed no stationary state at all. Unfortunately the attempts of these authors have not been successful; their method of series expansion proved to be powerless to control manyparticle problems. In the present paper, the writer wishes to settle this problem by showing rigorously that the wave equation for the two-electron problem, in particular the helium wave equation, has a very large number-even infinity if the nucleus is assumed to be infinitely heavy-of solutions corresponding to stationary states of the system. In particular the existence of the ground state solution is established. Our method is different from those adopted by the authors cited above. We shall not attempt, for the present, to find explicit expressions for the solutions and shall resort to a method based on the abstract theory of operators in Hilbert space. The essential part of our theory may be regarded as already completed in the previous paper(4), where it was shown that the Hamiltonian operator of every atom, molecule, or ion is essentially selfadjoint(5). This means that the closed Hamiltonianfl of such a system, which is uniquely determined by the given Hamiltonian as a formal differential operator, is self-adjoint in the strict sense, that is, that it has a complete set (discrete or continuous) of eigenfunctions. Further, it was shown that eigen-