Biorthonormal Bases in Hilbert Space

  • Luc Lathouwers


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.


Hilbert Space Schrodinger Equation Orthonormal System Secular Equation Finite Dimensional Case 
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Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Luc Lathouwers
    • 1
  1. 1.Dienst voor Teoretische en Wiskundige NatuurkundeRijksuniversitair Centrum AntwerpenAntwerpBelgium

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