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11: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation … ►With the normalization factor ${\left(c{h}_n\right)}^{-1/2}$ the ${\psi }_n$ are orthonormal in $L^2(ℝ,dx)$. … ►The orthonormal stationary states and corresponding eigenvalues are then of the form …The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►with an infinite set of orthonormal $L^2$ eigenfunctions …

12: 3.5 Quadrature

… ►The corresponding orthonormal polynomials $q_n(x)=p_n(x)/\sqrt{h_n}$ satisfy the recurrence relation … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►The monic version $p_n(x)$ and orthonormal version $q_n(x)$ of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by $k_n$ respectively $\sqrt{h_n}$, with $k_n$ and $h_n$ as in Table 18.3.1. … ►

Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions $p_n(x)$ and orthonormal versions $q_n(x)$ of the classical orthogonal polynomials.

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$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
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13: Bibliography K

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

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14: 18.9 Recurrence Relations and Derivatives

… ►They imply the recurrence coefficients for the orthonormal versions of the classical OP’s as well, see again §3.5(vi). …

15: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

Just below (33.14.9), the constraint described in the text “ when ,” was removed. In Equation (33.14.13), the constraint ${ϵ}_1,{ϵ}_2>0$ was added. In the line immediately below (33.14.13), it was clarified that $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ times a polynomial in $r/n$, instead of simply a polynomial in $r$. In Equation (33.14.14), a second equality was added which relates ${\varphi }_{n,\mathrm{\ell }}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system.

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