Conjugate eigenvalue problem

A short presentation of the unpublished Research Report No 217, by Osvaldo Goscinski, from the Quantum Chemistry Group at Uppsala University is made. The Report has the title Conjugate Eigenvalue Problems and the Theory of Upper and Lower Bounds . Some justification of its verbatim inclusion in this Volume in honour of Per-Olov Lowdin is made. It is essentially motivated by the attention that the theory presented there has received in the field of Generalized and Molecular Sturmians. Current work by John Avery and collaborators is alluded to. It is included as an Appendix. 

This is what is meant by the conjugate eigenvalue problem. The perturbation involved is the simple Coulomb operator, or parts of it, appearing in molecular physics. 

This is actually the work in question. The generalization to a many-electron conjugate eigenvalue problem was done in 1968 but the actual use of the name Sturmian was not done there. 

The author is grateful to Professsor John Avery for his interest in Sturmian functions and in the unpublished report from 1968. He is further indebted to him for a Latex translation of the original manuscript where the conjugate eigenvalue problem as well Sturmians are introduced. 

CONJUGATE EIGENVALUE PROBLEMS AND THE THEORY OF UPPER AND LOWER BOUNDS... 

These conjugate eigenvalue problems are of course, well known in the... 

The very interesting work of Joseph on the determination of the exact number of bound states of a given potential uses the conjugate eigenvalue problem for arbitrary one particle, N-dimensional potentials. It turns out that the conjugate problem is exactly soluble in several cases of interest, but for an arbitrary problem the techniques discussed in this paper, with approximate solutions, are needed. 

In combining the projection technique for upper and lower bounds with the conjugate eigenvalue problems, we are able to extend some of the results of Joseph, on the counting of the eigenvalues of given one-particle potentials, to arbitrary problems. Approximate techniques are of course needed. 

Goscinski O (1968) Conjugate eigenvalue problems and the theory of upper and lower bounds. Research Report 217, Quantum Chemistry Group, Uppsala University... 

O. Goscinski Conjugate Eigenvalue Problems and the Theory of Upper and Lower Bounds (Preprint No. 217, June 17, 1968). 

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