fixed points

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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. … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ …This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed $l$, do not form a complete set. Namely for fixed $l$ the infinite set labeled by $p$ describe only the $L^2$ bound states for that single $l$, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … ►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. …