# Hermite differential operator

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##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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, $n=0,1,2,\ldots$), with eigenvalues $\lambda_{n}=2n+1\in\boldsymbol{\sigma}_{p}$, for the differential operator
1.18.43 $f(x)=\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}\frac{{\mathrm{e}}^{-(x^{2}+y^{% 2})/2}}{{\pi}^{\frac{1}{2}}2^{n}n!}H_{n}\left(x\right)H_{n}\left(y\right)f(y)% \,\mathrm{d}y,$
##### 2: Bibliography R
• J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
• ##### 3: 18.27 $q$-Hahn Class
In the $q$-Hahn class OP’s the role of the operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative $\mathcal{D}_{q}$, as defined in (17.2.41). …
##### 4: 18.3 Definitions
###### §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 $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). …
##### 5: 18.38 Mathematical Applications
###### Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
The basic ideas of Gaussian quadrature, and their extensions to non-classical 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 so-called Casimir operator
##### 6: Bibliography K
• T. H. Koornwinder (2006) Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.
• T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
• T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.
• A. B. J. Kuijlaars and R. Milson (2015) Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, pp. 28–39.
• K. H. Kwon, L. L. Littlejohn, and G. J. Yoon (2006) Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.
• ##### 7: Errata
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 $\mathcal{V}_{a,b}\left(f(t)\right)=\int_{a}^{b}\left|f^{\prime}(t)\right|\,% \mathrm{d}t$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Juan Luis Varona on 2021-02-08

• Equation (1.4.34)
1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Tran Quoc Viet on 2020-08-11

• Subsection 1.16(vii)

Several changes have been made to

1. (i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$, $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ 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));

2. (ii)

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

3. (iii)

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

4. (iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

• ##### 8: 18.36 Miscellaneous Polynomials
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 self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).
##### 9: 18.39 Applications in the Physical Sciences
The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathcal{H}$, is a second order differential operator of the form … c) A Rational SUSY Potential argumentAnalogous to (18.39.7) the 3D Schrödinger operator is … The radial operator (18.39.28) …
##### 10: 1.17 Integral and Series Representations of the Dirac Delta
for all functions $\phi(x)$ that are continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$. … More generally, assume $\phi(x)$ is piecewise continuous (§1.4(ii)) when $x\in[-c,c]$ for any finite positive real value of $c$, and for each $a$, $\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$. … provided that $\phi(x)$ is continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ 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 co-ordinates) 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. …