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31: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being

L^{2}

and forming a complete set. … ►Thus the two missing quantum numbers corresponding to EOP’s of order

1

and

2

of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. … ►The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). …