When Per-Olov Löwdin solved the non-othogonality problem, in his Ph.D. thesis1 in 1947, he did this through an ingenious combination of mathematical tools and physical intuition. Like orthonormalisation, biorthonormalisation is a problem of both mathematical and physical importance. Covariant and contravariant representations, direct and reciprocal lattices, secular equations for non-orthogonal basis sets are just a few topics intimately related to the concept of biorthonormal bases. The full solution of the biorthonormality problem, i.e., an existence theorem for biorthonormalisation, has, to our knowledge, not been given. It is the intention of the present article to fill this gap. or this purpose we will use a theorem of R. Paley and N. Wiener and the symmetric orthonormalisation 1, thus following Per-Olov Löwdin’s example in combining mathematics and physics to solve a problem of interest to both fields.
KeywordsHilbert Space Schrodinger Equation Orthonormal System Secular Equation Finite Dimensional Case
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- 1.P.O. Löwdin, “A Theoretical Investigation into some Properties of Ionic Crystals”, Thesis, Uppsala 1948, Almqvist and Wiksells.Google Scholar
- 2.R. Paley and N. Wiener, “Fourier Transforms in the Complex Domain”, Am. Math. Soc. (1934) 100.Google Scholar
- 3.P.O. Löwdin, “Advances in Quantum Chemistry V”, Academic Press (1970).Google Scholar
- 4.S. Banach, “Theorie des opérations linéaires”, Monografje Matematyczne (1932) 80 th. 5.Google Scholar
- 6.B. Sz.-Nagy, Duke Math. J. 14 (1947) 975.Google Scholar
- 7.I. Singer, “Bases in Banach Spaces I”, Springer (1970), this book contains an extensive bibliography on basis problems upto 1970.Google Scholar
- 8.J. von Neumann, “Mathematical Foundations of Quantum Mechanics”, Princeton U.P. (1955) 184.Google Scholar
- 10.J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498.Google Scholar
- 13.L. Lathouwers, accepted for publ. in Intern. J. Quant. Chem.Google Scholar
- 15.F. Riesz, Comptes Rendus 144 (1934) 34.Google Scholar
- 16.F. Riesz and B. Sz.-Nagy, “Functional Analysis” F. Unger Publ. Co., New York.Google Scholar
- 18.L. Lathouwers, submitted for publ. in Intern. J. Quant. Chem.Google Scholar