…

►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of

Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original

*AMS 55 NBS Handbook of Mathematical Functions*.
…

…

►The sum of the kinetic and potential energies give the quantum

*Hamiltonian*, or energy

operator; often also referred to as a

*Schrödinger operator*.
…
…

►
###### Example 2: Radial 3D Schrödinger operators, including the Coulomb potential

…

► See §

18.39 for discussion of

Schrödinger equations and

operators.
…

…

►The fundamental quantum

*Schrödinger operator*, also called the Hamiltonian,

$\mathscr{H}$, is a second order differential

operator of the form
…

►
######
§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

►Analogous to (

18.39.7) the 3D

Schrödinger operator is
…

►noting that the

${\psi}_{p,l}(r)$ are real, follows from the fact that the

Schrödinger operator of (

18.39.28) is self-adjoint, or from the direct derivation of

Dunkl (2003).
…

►Here tridiagonal representations of simple

Schrödinger operators play a similar role.
…

…

►Completeness and orthogonality follow from the self-adjointness of the corresponding

Schrödinger operator,

Gómez-Ullate and Milson (2014),

Marquette and Quesne (2013).

…

►Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent

Schrödinger equation.
…

►A further

operator, the so-called

*Casimir operator*
…

►
###### Dunkl Type Operators and Nonsymmetric
Orthogonal Polynomials

…

►
…

►Hermite EOP’s appear in solutions of a rationally modified

Schrödinger equation in §

18.39.
…

…

►We use vector notation

$[\mathbf{s},\mathbf{t},\mathbf{u}]$ (respective scalar

$(s,t,u)$) for any one of the three spin

operators (respective spin values).
…

►
$${\mathit{H}}_{s}\mathrm{\Psi}(\mathbf{x})\equiv (-2\mathbf{s}\cdot \mathbf{t}-(2/a)\mathbf{s}\cdot \mathbf{u})\mathrm{\Psi}(\mathbf{x})={h}_{s}\mathrm{\Psi}(\mathbf{x}),$$

…

►The

operators
${\mathbf{J}}^{2}$ and

${\mathit{H}}_{s}$ admit separation of variables in

${z}_{1},{z}_{2}$, leading to the following factorization of the eigenfunction

$\mathrm{\Psi}(\mathbf{x})$:
…

►Heun functions appear in the theory of black holes (

Kerr (1963),

Teukolsky (1972),

Chandrasekhar (1984),

Suzuki et al. (1998),

Kalnins et al. (2000)), lattice systems in statistical mechanics (

Joyce (1973, 1994)), dislocation theory (

Lay and Slavyanov (1999)), and solution of the

Schrödinger equation of quantum mechanics (

Bay et al. (1997),

Tolstikhin and Matsuzawa (2001), and

Hall et al. (2010)).
…

…

►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.
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.
…

►
Equation (3.3.34)
In the online version, the leading divided difference operators were previously
omitted from these formulas, due to programming error.

*Reported by Nico Temme on 2021-06-01*

…

►
Subsections 1.15(vi), 1.15(vii), 2.6(iii)
A number of changes were made with regard to fractional integrals and derivatives.
In §1.15(vi) a reference to Miller and Ross (1993) was added,
the fractional integral operator of order $\alpha $ was more precisely identified as the
*Riemann-Liouville* fractional integral operator of order $\alpha $, and a paragraph was added below
(1.15.50) to generalize (1.15.47).
In §1.15(vii) the sentence defining the fractional derivative was clarified.
In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator
was made consistent with §1.15(vi).

…