…
►Writing
Hermite’s
differential equation (see Tables
18.3.1 and
18.8.1) in the form above, the eigenfunctions are
${\mathrm{e}}^{{x}^{2}/2}{H}_{n}\left(x\right)$ (
${H}_{n}$ a
Hermite polynomial,
$n=0,1,2,\mathrm{\dots}$), with eigenvalues
${\lambda}_{n}=2n+1\in {\bm{\sigma}}_{p}$, for the
differential operator
…
►
1.18.43
$$f(x)=\sum _{n=0}^{\mathrm{\infty}}{\int}_{\mathrm{\infty}}^{\mathrm{\infty}}\frac{{\mathrm{e}}^{({x}^{2}+{y}^{2})/2}}{{\pi}^{\frac{1}{2}}{2}^{n}n!}{H}_{n}\left(x\right){H}_{n}\left(y\right)f(y)dy,$$
…
…
►In the
$q$Hahn class OP’s the role of the
operator
$d/dx$ in the Jacobi, Laguerre, and
Hermite cases is played by the
$q$derivative
${\mathcal{D}}_{q}$, as defined in (
17.2.41).
…
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and
Hermite polynomials.
…
►
1.
As eigenfunctions of second order differential operators
(Bochner’s theorem, Bochner (1929)).
See the differential equations
$A(x){p}_{n}^{\prime \prime}(x)+B(x){p}_{n}^{\prime}(x)+{\lambda}_{n}{p}_{n}(x)=0$,
in Table 18.8.1.
…
►Table
18.3.1 provides the
traditional definitions of Jacobi, Laguerre, and
Hermite polynomials via orthogonality and standardization (§§
18.2(i) and
18.2(iii)).
…
►For finite power series of the Jacobi, ultraspherical, Laguerre, and
Hermite polynomials, see §
18.5(iii) (in powers of
$x1$ for Jacobi polynomials, in powers of
$x$ for the other cases).
…
…
►
Quadrature “Extended” to PseudoSpectral (DVR) Representations of Operators in One and Many Dimensions
►The basic ideas of Gaussian quadrature, and their extensions to nonclassical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to
discrete variable representations, or DVRs, of Sturm–Liouville and other
differential operators.
…
►A further
operator, the socalled
Casimir operator
…
►
Dunkl Type Operators and Nonsymmetric
Orthogonal Polynomials
…
►
…
…
►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order
differential operators and eigenfunction expansions.
…
►The specific updates to Chapter
1 include the addition of an entirely new subsection §
1.18 entitled “Linear Second Order
Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order
differential operators.
…
►
Equation (2.3.6)
2.3.6
$${\mathcal{V}}_{a,b}\left(f(t)\right)={\int}_{a}^{b}\left{f}^{\prime}(t)\rightdt$$
The integrand has been corrected so that the absolute value does not include the differential.
Reported by Juan Luis Varona on 20210208
…
►
Equation (1.4.34)
1.4.34
$${\mathcal{V}}_{a,b}\left(f\right)={\int}_{a}^{b}\left{f}^{\prime}(x)\rightdx$$
The integrand has been corrected so that the absolute value does not include the differential.
Reported by Tran Quoc Viet on 20200811
…
►
Subsection 1.16(vii)
Several changes have been made to

(i)
make consistent use of the Fourier transform notations $\mathcal{F}\left(f\right)$,
$\mathcal{F}\left(\varphi \right)$ and $\mathcal{F}\left(u\right)$ where $f$ is a function of one
real variable, $\varphi $ is a test function of $n$ variables associated with
tempered distributions, and $u$ is a tempered distribution
(see (1.14.1), (1.16.29) and (1.16.35));

(ii)
introduce the partial differential operator
$\mathbf{D}$ in (1.16.30);

(iii)
clarify the definition (1.16.32) of the partial differential
operator
$P(\mathbf{D})$; and

(iv)
…
…
►They are related to
Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers.
…
►
Type III ${X}_{2}$Hermite EOP’s
►Hermite EOP’s are defined in terms of classical
Hermite OP’s.
The type III
${X}_{2}$
Hermite EOP’s, missing polynomial orders
$1$ and
$2$, are the complete set of polynomials, with real coefficients and defined explicitly as
…
►Completeness and orthogonality follow from the selfadjointness of the corresponding Schrödinger
operator,
GómezUllate and Milson (2014),
Marquette and Quesne (2013).
…
►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of selfadjoint second order
differential operators is overviewed in §
1.18.
…
►The fundamental quantum
Schrödinger operator, also called the Hamiltonian,
$\mathscr{H}$, is a second order
differential operator of the form
…
►c) A Rational SUSY Potential
argument
…
►Analogous to (
18.39.7) the 3D Schrödinger
operator is
…
►The radial
operator (
18.39.28)
…
…
►for all functions
$\varphi (x)$ that are continuous when
$x\in (\mathrm{\infty},\mathrm{\infty})$, and for each
$a$,
${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{n{(xa)}^{2}}\varphi (x)dx$ converges absolutely for all sufficiently large values of
$n$.
…
►More generally, assume
$\varphi (x)$ is piecewise continuous (§
1.4(ii)) when
$x\in [c,c]$ for any finite positive real value of
$c$, and for each
$a$,
${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{n{(xa)}^{2}}\varphi (x)dx$ converges absolutely for all sufficiently large values of
$n$.
…
►provided that
$\varphi (x)$ is continuous when
$x\in (\mathrm{\infty},\mathrm{\infty})$, and for each
$a$,
${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{n{(xa)}^{2}}\varphi (x)dx$ converges absolutely for all sufficiently large values of
$n$ (as in the case of (
1.17.6)).
…
►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these
non${L}^{2}$ improper eigenfunctions of the
differential operators (with derivatives respect to spatial coordinates) which generate them; the normalizations (
1.17.12_1) and (
1.17.12_2) are explicitly derived in
Friedman (1990, Ch. 4), the others follow similarly.
…
►
Hermite Polynomials (§18.3)
…