resolvent

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1: Viewing DLMF Interactive 3D Graphics

… ►Until these issues are resolved we cannot guarantee that the DLMF WebGL visualizations can be viewed in Internet Explorer. …

2: 1.11 Zeros of Polynomials

… ►For the roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ of $g(w)=0$ and the roots ${\theta }_1,{\theta }_2,{\theta }_3$ of the resolvent cubic equation … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►this being a matrix element of the resolvent $F(T)={\left(z-T\right)}^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). … ►

1.18.66 $$⟨{\left(z-T\right)}^{-1}f,f⟩=\sum _{{𝝈}_p}\frac{{\left|\widehat{f}({\lambda }_n)\right|}^2}{z-{\lambda }_n}+{\int }_{{𝝈}_c}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$ $f\in L^2\left(X\right)$, $z\notin 𝝈$.

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral, $\notin$: not an element of, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(viii), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E66

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

… ► … ►In unusual cases $N=\mathrm{\infty }$, even for all $\mathrm{\ell }$, such as in the case of the Schrödinger–Coulomb problem ($V=-r^{-1}$) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at $\lambda =0$, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). … ►The resolvent set $\rho (T)$ consists of all $z\in ℂ$ such that (i) $z-T$ is injective, (ii) $ℛ(z-T)$ is dense in $V$, (iii) the resolvent ${\left(z-T\right)}^{-1}$ is bounded. …

4: 28.14 Fourier Series

… ►Ambiguities in sign are resolved by (28.14.9) when $q=0$, and by continuity for other values of $q$. …

5: 20.1 Special Notation

… ► ►►►

$m$, $n$	integers.
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$q^{\alpha }$	${\mathrm{e}}^{\mathrm{i}\alpha \pi \tau }$ for $\alpha \in ℝ$ (resolving issues of choice of branch).
…

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6: 4.21 Identities

… ►In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities. …

7: 28.4 Fourier Series

… ►Ambiguities in sign are resolved by (28.4.13)–(28.4.16) when $q=0$, and by continuity for the other values of $q$. …

8: 28.31 Equations of Whittaker–Hill and Ince

… ►ambiguities in sign being resolved by requiring $C_p^m(x,\xi )$ and $S_p^m{}_{}{}^{\prime }(x,\xi )$ to be continuous functions of $x$ and positive when $x=0$. …

9: 1.6 Vectors and Vector-Valued Functions

… ►Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over). …

10: 28.2 Definitions and Basic Properties

… ►the ambiguity of sign being resolved by (28.2.29) when $q=0$ and by continuity for the other values of $q$. …