§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

A linear operator $T$ on a (complex) linear vector space $V$ is a map $T:V\to V$ such that

1.18.21		$$T(\alpha v+\beta w)=\alpha Tv+\beta Tw,$$
$v,w\in V$, $\alpha ,\beta \in ℂ$.
ⓘ Symbols: $ℂ$: complex plane, $\in$: element of and $w$: variable Permalink: http://dlmf.nist.gov/1.18.E21 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

In the following let $V$ be a Hilbert space. A linear operator $T$ on $V$ is bounded with norm $\Vert T\Vert$ if

1.18.22
ⓘ Symbols: $\in$: element of, $\equiv$: equals by definition, $\Vert 𝐀\Vert$: matrix norm, $sup$: least upper bound (supremum) and $\Vert 𝐯\Vert$: vector norm ($l$²) Permalink: http://dlmf.nist.gov/1.18.E22 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

More generally, a linear operator $T$ on $V$ needs not be defined on all of $V$, but only on a linear subspace $𝒟(T)$ of $V$ which is called the domain of $T$. Then $T:𝒟(T)\to V$ is a linear map. Assume that $𝒟(T)$ is dense in $V$, i.e., for each $v\in V$ there is a sequence $\{v_n\}$ in $𝒟(T)$ such that $\Vert v_n-v\Vert \to 0$ as $n\to \mathrm{\infty }$. If ${sup}_{v\in 𝒟(T),\Vert v\Vert =1}\Vert Tv\Vert$ is finite then $T$ is bounded, and $T$ extends uniquely to a bounded linear operator on $V$. If the supremum is $\mathrm{\infty }$, then $T$ is an unbounded linear operator on $V$.

If $T$ is a bounded linear operator on $V$ then its adjoint is the bounded linear operator $T^{\ast }$ such that, for $v,w\in V$,

1.18.23		$$⟨Tv,w⟩=⟨v,T^{\ast }w⟩.$$
ⓘ Symbols: ${𝐀}^{\ast }$: adjoint of matrix, $⟨𝐮,𝐯⟩$: inner product over vectors and $w$: variable Referenced by: §1.18(iii) Permalink: http://dlmf.nist.gov/1.18.E23 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

The operator $T$ is called self-adjoint if $T^{\ast }=T$, and referred to as symmetric if (1.18.23) holds for $v,w$ in the dense domain $𝒟(T)$ of $T$. There is also a notion of self-adjointness for unbounded operators, see §1.18(ix). One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner.

An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions

1.18.24		$$T{u}_{\lambda }=\lambda u_{\lambda },$$
ⓘ Permalink: http://dlmf.nist.gov/1.18.E24 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

$u_{\lambda }\in 𝒟(T)$, corresponding to distinct eigenvalues, are orthogonal: i.e., $⟨u_{\lambda },u_{{\lambda }^{\prime }}⟩=0$, for $\lambda \ne {\lambda }^{\prime }$. If an eigenvalue has multiplicity $>1$, the eigenfunctions may always be orthogonalized in this degenerate sub-space.

Focus is now placed on second order differential operators as these are the subject of the remainder of §1.18.

Consider the second order differential operator acting on real functions of $x$ in the finite interval $[a,b]\subset ℝ$

1.18.25		$$T=\frac{d^2}{{dx}^2},$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ Permalink: http://dlmf.nist.gov/1.18.E25 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

and functions $f(x),g(x)\in C^2(a,b)$, assumed real for the moment. The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. We integrate by parts twice giving:

1.18.26		$${\int }_a^b{f}^{\prime \prime }(x)g(x)dx={f^{\prime }(x)g(x)|}_a^b-{f(x)g^{\prime }(x)|}_a^b+{\int }_a^{b}f(x)g^{\prime \prime }(x)dx.$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral Referenced by: §1.18(iii), §1.18(iii) Permalink: http://dlmf.nist.gov/1.18.E26 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

Ignoring the boundary value terms it follows that

1.18.27		$$T^{\ast }=T=\frac{d^2}{{dx}^2},$$
ⓘ Symbols: ${𝐀}^{\ast }$: adjoint of matrix and $\frac{df}{dx}$: derivative of $f$ with respect to $x$ Permalink: http://dlmf.nist.gov/1.18.E27 Encodings: TeX, pMML, png See also: Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

and thus $T$ is said to be formally self adjoint.

For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$.

This question may be rephrased by asking: do $f(x)$ and $g(x)$ satisfy the same boundary conditions which are needed to fully specify the solutions of a second order linear differential equation? A simple example is the choice $f(a)=f(b)=0$, and $g(a)=g(b)=0$, this being only one of many. This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that $𝒟(T)=𝒟(T^{\ast })$, as $f(x)$ and $g(x)$ satisfy the same boundary conditions and thus define the same domains. Thus $T$ is indeed self adjoint.

Other choices of boundary conditions, identical for $f(x)$ and $g(x)$, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of $T$. The nature of these extensions for unbounded intervals such as $[0,\mathrm{\infty })$, and unbounded operators on them, are the subject of §1.18(ix).

Let $T$ be a self-adjoint extension of differential operator $ℒ$ of the form (1.18.28) and assume $T$ has a complete set of $L^2$ eigenfunctions, ${\left\{{\varphi }_{{\lambda }_n}(x)\right\}}_{n=0}^{\mathrm{\infty }}$ , $x\in X=[a,b]$ this latter being an appropriate sub-set of $ℝ$, or, in some cases $X=ℝ$ itself, with real eigenvalues ${\lambda }_n$. These eigenvalues will be assumed distinct, i.e., of unit multiplicity, unless otherwise stated. The point, or discrete spectrum of $T$ is then given by ${𝝈}_p=\{{\lambda }_0,{\lambda }_1,\mathrm{\dots }\}$. The eigenfunctions form a complete orthogonal basis in $L^2\left(X\right)$, and we can take the basis as orthonormal:

1.18.29		$${\int }_a^b{\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_m}(x)}dx={\delta }_{n,m},$$
ⓘ Symbols: ${\delta }_{j,k}$: Kronecker delta, $\overline{z}$: complex conjugate, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §1.18(v), §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E29 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

and completeness implies

1.18.30		$$\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_n}(y)}=\delta \left(x-y\right).$$
ⓘ Symbols: $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate and $n$: nonnegative integer Referenced by: §1.18(v), §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E30 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

Now formulas (1.18.13)–(1.18.20) apply. For $f(x)\in C(X)\cap L^2\left(X\right)\cap 𝒟(T)$, $f(x)$ has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19),

1.18.31		$$f(x)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{{\lambda }_n}(x){\int }_a^{b}f(y)\overline{{\varphi }_{{\lambda }_n}(y)}dy=\sum _{n=0}^{\mathrm{\infty }}\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x)$$
ⓘ Symbols: $\overline{z}$: complex conjugate, $dx$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.18(v) Permalink: http://dlmf.nist.gov/1.18.E31 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

where

1.18.32		$$\widehat{f}({\lambda }_n)=⟨f,{\varphi }_{{\lambda }_n}⟩.$$
ⓘ Symbols: $⟨f,g⟩$: inner product over functions and $n$: nonnegative integer Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E32 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

Further

1.18.33		$${\int }_a^b{\left|f(x)\right|}^{2}dx=\sum _{n=0}^{\mathrm{\infty }}{\left|\widehat{f}({\lambda }_n)\right|}^2.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$ Referenced by: §1.18(vi) Permalink: http://dlmf.nist.gov/1.18.E33 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

Spectral expansions of $T$, and of functions $F(T)$ of $T$, these being expansions of $T$ and $F(T)$ in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow:

1.18.34		$$(Tf)(x)=\sum _{n=0}^{\mathrm{\infty }}{\lambda }_n\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x)={\int }_a^b\left(\sum _{n=0}^{\mathrm{\infty }}{\lambda }_n{\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_n}(y)}\right)f(y)dy,$$
ⓘ Symbols: $\overline{z}$: complex conjugate, $dx$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.18(vi) Permalink: http://dlmf.nist.gov/1.18.E34 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

1.18.35		$$(F(T)f)(x)={\int }_a^b\left(\sum _{n=0}^{\mathrm{\infty }}F({\lambda }_n){\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_n}(y)}\right)f(y)dy.$$
ⓘ Symbols: $\overline{z}$: complex conjugate, $dx$: differential of $x$, $\int$: integral and $n$: nonnegative integer Referenced by: §1.18(vi) Permalink: http://dlmf.nist.gov/1.18.E35 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

Possible eigenfunctions of $-\frac{d^2}{{dx}^2}$ being $\sin \left(kx\right)$, $\cos \left(kx\right)$, ${\mathrm{e}}^{\pm \mathrm{i}kx}$, consider three cases, which illustrate the importance of boundary conditions.

Case 1: $\varphi (0)=\varphi (\pi )=0$. The $L^2$ normalized eigenfunctions on $[0,\pi ]$ are

1.18.36		$${\varphi }_{\sin }(n,x)=\sqrt{\frac{2}{\pi }}\sin \left(nx\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $(a,b)$: open interval, $\sin z$: sine function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E36 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

with ${\lambda }_n=n^2$, $n=1,2,3,\mathrm{\dots }$.

Case 2: ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )=0$. The $L^2$ normalized eigenfunctions on $[0,\pi ]$ are

1.18.37		$${\varphi }_{\cos }(0,x)=\frac{1}{\sqrt{\pi }},{\varphi }_{\cos }(n,x)=\sqrt{\frac{2}{\pi }}\cos \left(nx\right),$$
$n=1,2,3\mathrm{\dots }$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $(a,b)$: open interval and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E37 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

with ${\lambda }_n=n^2$, $n=0,1,2,\mathrm{\dots }$.

Case 3: Periodic Boundary Conditions: $\varphi (0)=\varphi (\pi )$ and ${\varphi }^{\prime }(0)={\varphi }^{\prime }(\pi )$. The $L^2$ normalized eigenfunctions on $[0,\pi ]$ are

1.18.38		$${\varphi }_{\exp }(\pm n,x)=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{\pm 2\mathrm{i}nx},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E38 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

with ${\lambda }_{\pm n}=4n^2$, $n=0,1,2,\mathrm{\dots }$, with all eigenvalues, for $\left|n\right|>0$, having multiplicity two, as changing the sign of $n$ changes the eigenfunction but not the eigenvalue, and multiplicity one for $n=0$. Letting $n$ run from $-\mathrm{\infty }$ to $\mathrm{\infty }$ this multiplicity change is automatically included:

1.18.39		$$\delta \left(x-y\right)=\frac{1}{\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}n(x-y)}.$$
ⓘ Symbols: $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E39 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

This may be compared to (1.17.21), the resulting Fourier, or eigenfunction, expansion

1.18.40		$$f(x)=\frac{1}{\sqrt{\pi }}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}nx}\widehat{f}({\lambda }_n),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E40 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

where $\widehat{f}({\lambda }_n)$= $\frac{1}{\sqrt{\pi }}{\int }_0^{\pi }f(y){\mathrm{e}}^{-2\mathrm{i}ny}dy=⟨f,{\varphi }_{\exp }(n)⟩$, being that of (1.8.3) and (1.8.4). The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above.

The space $X$ is now the full real line, $(-\mathrm{\infty },\mathrm{\infty })$. 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 ${\lambda }_n=2n+1\in {𝝈}_p$, for the differential operator

1.18.41		$${ℒ}^{\mathrm{Hermite}}=-\frac{d^2}{{dx}^2}+x^2,$$
$x\in X$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $\in$: element of Permalink: http://dlmf.nist.gov/1.18.E41 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

Applying equations (1.18.29) and (1.18.30), the complete set of normalized eigenfunctions being

1.18.42		$${\varphi }_n(x)=\frac{1}{\sqrt{{\pi }^{\frac{1}{2}}{2}^{n}n!}}{\mathrm{e}}^{-x^2/2}{H}_n\left(x\right),$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E42 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

(1.18.31) becomes

1.18.43		$$f(x)=\sum _{n=0}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-(x^2+y^2)/2}}{{\pi }^{\frac{1}{2}}{2}^{n}n!}{H}_n\left(x\right)H_n\left(y\right)f(y)dy,$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E43 Encodings: TeX, pMML, png See also: Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

for $f(x)\in L^2$ and piece-wise continuous, with convergence as discussed in §1.18(ii).

See Titchmarsh (1962a, pp. 73–75). The implicit boundary conditions taken here are that the ${\varphi }_n(x)$ and ${\varphi }_n^{\prime }(x)$ vanish as $x\to \pm \mathrm{\infty }$, which in this case is equivalent to requiring ${\varphi }_n(x)\in L^2\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point.

Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. Let $T$ be the self adjoint extension of a formally self-adjoint differential operator $ℒ$ of the form (1.18.28) on an unbounded interval $X\subset ℝ$, which we will take as $X=[0,+\mathrm{\infty })$, and assume that $q(x)\to 0$ monotonically as $x\to \mathrm{\infty }$, and that the eigenfunctions are non-vanishing but bounded in this same limit. Assume $T$ has no point spectrum, i.e., $T$ has no eigenfunctions in $L^2\left(X\right)$, then the spectrum $𝝈$ of $T$ consists only of a continuous spectrum, referred to as ${𝝈}_c$. In this subsection it is assumed that ${𝝈}_c=[0,\mathrm{\infty })$. This will be generalized, along with the choice of $X$, in §1.18(vii).

Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have

1.18.44		$${\int }_0^{\mathrm{\infty }}{\varphi }_{\lambda }(x)\overline{{\varphi }_{{\lambda }^{\prime }}(x)}dx=\delta \left(\lambda -{\lambda }^{\prime }\right),$$
$\lambda ,{\lambda }^{\prime }\in {𝝈}_c$,
ⓘ Symbols: $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $dx$: differential of $x$, $\in$: element of and $\int$: integral Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E44 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

and completeness relation

1.18.45		$$\delta \left(x-y\right)={\int }_0^{\mathrm{\infty }}{\varphi }_{\lambda }(x)\overline{{\varphi }_{\lambda }(y)}d\lambda .$$
ⓘ Symbols: $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $dx$: differential of $x$ and $\int$: integral Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E45 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

See Friedman (1990, pp. 233–252) for elementary discussions of both equations and the normalization process; and also the references in §1.18(ix).

Now formulas (1.18.13)–(1.18.20) apply. For $f(x)\in C(X)\cap L^2\left(X\right)\cap 𝒟(T)$, $f(x)$ has the eigenfunction expansion, analogous to that of (1.18.33),

1.18.46		$$f(x)={\int }_0^{\mathrm{\infty }}{\varphi }_{\lambda }(x)\widehat{f}(\lambda )d\lambda ,$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.18.E46 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

where

1.18.47		$$\widehat{f}(\lambda )=⟨f,{\varphi }_{\lambda }⟩.$$
ⓘ Symbols: $⟨f,g⟩$: inner product over functions Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E47 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

Further,

1.18.48		$${\int }_0^{\mathrm{\infty }}{\left|f(x)\right|}^{2}dx={\int }_0^{\mathrm{\infty }}{\left|\widehat{f}(\lambda )\right|}^{2}d\lambda .$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $\left|x\right|$: absolute value of $x$ Permalink: http://dlmf.nist.gov/1.18.E48 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

The analog of (1.18.34) is

1.18.49		$$(Tf)(x)={\int }_0^{\mathrm{\infty }}\left({\int }_0^{\mathrm{\infty }}\lambda {\varphi }_{\lambda }(x)\overline{{\varphi }_{\lambda }(y)}d\lambda \right)f(y)dy,$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E49 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

and that of (1.18.35) is

1.18.50		$$(F(T)f)(x)={\int }_0^{\mathrm{\infty }}\left({\int }_0^{\mathrm{\infty }}F(\lambda ){\varphi }_{\lambda }(x)\overline{{\varphi }_{\lambda }(y)}d\lambda \right)f(y)dy.$$
ⓘ Symbols: $dx$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.18.E50 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

This implies

1.18.51		$$⟨F(T)f,f⟩={\int }_0^{\mathrm{\infty }}F(\lambda ){\left|\widehat{f}(\lambda )\right|}^{2}d\lambda .$$
ⓘ Symbols: $dx$: differential of $x$, $⟨f,g⟩$: inner product over functions, $\int$: integral and $\left|x\right|$: absolute value of $x$ Permalink: http://dlmf.nist.gov/1.18.E51 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

In particular, this holds for $F(\lambda )={\left(z-\lambda \right)}^{-1}$, $z\in ℂ\setminus {𝝈}_c$

1.18.52		$$⟨{\left(z-T\right)}^{-1}f,f⟩={\int }_{𝝈}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$
$f\in L^2\left(X\right)$, $z\notin {𝝈}_c$,
ⓘ 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 and $\left|x\right|$: absolute value of $x$ Referenced by: §1.18(vi), §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E52 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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). Then

1.18.53		$$\underset{\nu \to 0+}{lim}\frac{1}{2\pi \mathrm{i}}\left(⟨{\left(\mu -\mathrm{i}\nu -T\right)}^{-1}f,f⟩-⟨{\left(\mu +\mathrm{i}\nu -T\right)}^{-1}f,f⟩\right)$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $⟨f,g⟩$: inner product over functions Referenced by: §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E53 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

gives ${\left|\widehat{f}(\mu )\right|}^2$ for $\mu \in {𝝈}_c$, and $0$ otherwise. This is the discontinuity across the branch cut in (1.18.52) ${𝝈}_c\subset ℝ$, from $z$ below to above the cut, divided by $2\pi \mathrm{i}$. See Newton (2002, §§7.1 and 7.3).