General form of the quantum-defect theory

Abstract

The quantum-defect-theory (QDT) treatment of an electron in the Coulomb field surrounding an ionic core is recast in a form largely independent of field characteristics and thus applicable, e.g., to square wells or to the Morse fields of diatomic molecules. The reformulation parallels Seaton's classification of alternative Coulomb-field wave functions, and makes it applicable to other fields. Wronskians of alternative pairs of base functions have an important role in the theory. For electron energies $\ensuremath{\epsilon}<0$ in a Coulomb field these Wronskians reduce to trigonometric functions of $\ensuremath{\nu}={(\ensuremath{-}2\ensuremath{\epsilon})}^{\ensuremath{-}\frac{1}{2}}$, familiar in the QDT; other fields lead to trigonometric functions of different arguments. Quantum defects are generalized to eigenvalues of a reaction matrix, as in Seaton's work, but this matrix can now be calculated even below threshold energies with the introduction of a "smooth" Green's function appropriate to QDT applications.