A method is presented for proving the existence and calculating lower bounds for critiml energies of the Hartrec equation for rhe helium atom. Our idea is to introduce an "energy" scalar product and use it to approximate the fourth order term in the potential by n smaller second order term. Rigorous lower bounds are then obtained from nn nssociared linwr operator containing il trial vector. By a suitable choice of the trial vecror WC are able to show that the minimal solution exists and is in fact a pointwise positive solution.
Recently, Reeken
[I] demonstrated the existence of a pointwise positive eigenfunction for the radial Hartree equation of the helium atom. Part of his analysis extends the theory of ref. [2] to unbounded regions. Several interesting articles on the mathematical properties of the Hartree equation have already appeared since Reeken's work [3-61. All of these results are primarily concerned with the properties of eigenfunctions in terms of the eigenvalue parameter. However, it is well known that in nonlinear problems the eigenvalue is usually not equal to the associated critical energy (see. for example, ref. [7]). In this article, we are concerned with the existence of the lowest critical energy of the Hartree equation for helium and the determination of upper and lower bounds to it, a problem which goes bacl: to Wilson and Lindsay [S] _ Once the existence has been established,