# §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

###### Contents

A survey is given of the formal spectral theory of second order differential operators, typical results being presented in §1.18(i) through §1.18(viii). The various types of spectra and the corresponding eigenfunction expansions are illustrated by examples. These are based on the Liouville normal form of (1.13.29). A more precise mathematical discussion then follows in §1.18(ix).

## §1.18(i) Hilbert spaces

A complex linear vector space $V$ is called an inner product space if an inner product $\left\langle u,v\right\rangle\in\mathbb{C}$ is defined for all $u,v\in V$ with the properties: (i) $\left\langle u,v\right\rangle$ is complex linear in $u$; (ii) $\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$; (iii) $\left\langle v,v\right\rangle\geq 0$; (iv) if $\left\langle v,v\right\rangle=0$ then $v=0$. With norm defined by

 1.18.1 $\left\|{v}\right\|=\sqrt{\left\langle v,v\right\rangle},$ ⓘ Symbols: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: inner product over vectors and $\left\|{\NVar{\mathbf{v}}}\right\|$: vector norm ($l$²) Referenced by: Erratum (V1.2.0) Section 1.18 Permalink: http://dlmf.nist.gov/1.18.E1 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

$V$ becomes a normed linear vector space. If $\left\|{v}\right\|=1$ then $v$ is normalized. Two elements $u$ and $v$ in $V$ are orthogonal if $\left\langle u,v\right\rangle=0$. A (finite or countably infinite, generalizing the definition of (1.2.40)) set $\{v_{n}\}$ is an orthonormal set if the $v_{n}$ are normalized and pairwise orthogonal.

An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_{n}\}$ in $V$ (i.e., $\lim_{m,n\to\infty}\left\|{v_{m}-v_{n}}\right\|=0$) converges in norm to some $v\in V$, i.e., $\lim_{n\to\infty}\left\|{v-v_{n}}\right\|=0$. For an orthonormal set $\{v_{n}\}$ in a Hilbert space $V$ Bessel’s inequality holds:

 1.18.2 $\sum_{n}{\left|c_{n}\right|}^{2}\leq{\left\|{v}\right\|}^{2},$

where $v\in V$ and

 1.18.3 $c_{n}=\left\langle v,v_{n}\right\rangle.$ ⓘ Symbols: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: inner product over vectors and $n$: nonnegative integer Referenced by: §1.18(i) Permalink: http://dlmf.nist.gov/1.18.E3 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

A Hilbert space $V$ is separable if there is an (at most countably infinite) orthonormal set $\{v_{n}\}$ in $V$ such that for every $v\in V$

 1.18.4 $\sum_{n}{\left|c_{n}\right|}^{2}={\left\|{v}\right\|}^{2},$ ⓘ Symbols: $\left\|{\NVar{\mathbf{v}}}\right\|$: vector norm ($l$²), $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Referenced by: §1.18(i) Permalink: http://dlmf.nist.gov/1.18.E4 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

where $c_{n}$ is given by (1.18.3). Such orthonormal sets are called complete. By (1.18.4)

 1.18.5 $\sum_{n=0}^{\infty}{\left|c_{n}\right|}^{2}<\infty.$ ⓘ Symbols: $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Referenced by: §1.18(i) Permalink: http://dlmf.nist.gov/1.18.E5 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

Conversely, if complex numbers $c_{n}$ satisfy (1.18.5) then there is a unique $v\in V$ such that (1.18.3) holds and $v$ can be given by

 1.18.6 $v=\sum_{n=0}^{\infty}c_{n}v_{n},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E6 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

where the infinite sum means convergence in norm,

 1.18.7 $\lim_{N\to\infty}\left\|{v-\sum_{n=0}^{N}c_{n}v_{n}}\right\|=0.$ ⓘ Symbols: $\left\|{\NVar{\mathbf{v}}}\right\|$: vector norm ($l$²) and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E7 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

The standard example of an (infinite dimensional) separable Hilbert space is the space $\ell^{2}$ with elements $v=(c_{0},c_{1},c_{2},\ldots)$ such that

 1.18.8 ${\left\|{v}\right\|}^{2}=\sum_{n=0}^{\infty}{\left|c_{n}\right|}^{2}<\infty.$

The inner product of $v$ and $w=(d_{0},d_{1},d_{2},\ldots)$ is

 1.18.9 $\left\langle v,w\right\rangle=\sum_{n=0}^{\infty}c_{n}\overline{d_{n}}.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: inner product over vectors, $w$: variable and $n$: nonnegative integer Referenced by: §1.18(ii) Permalink: http://dlmf.nist.gov/1.18.E9 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

Every infinite dimensional separable Hilbert space $V$ can be made isomorphic to $\ell^{2}$ by choosing a complete orthonormal set $\left\{v_{n}\right\}_{n=0}^{\infty}$ in $V$. Then an isomorphism is given by

 1.18.10 $\sum_{n=0}^{\infty}c_{n}v_{n}\mapsto(c_{0},c_{1},c_{2},\ldots)\colon V\to\ell^% {2}.$ ⓘ Symbols: $\ell$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E10 Encodings: TeX, pMML, png See also: Annotations for §1.18(i), §1.18 and Ch.1

General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82).

## §1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$

Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $\mathbb{R}$. For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$,

 1.18.11 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}\alpha(x)<\infty.$

Functions $f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)$ for which $\left\langle f-g,f-g\right\rangle=0$ are identified with each other. The space $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ becomes a separable Hilbert space with inner product

 1.18.12 $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)\overline{g(x)}\,\mathrm{d}% \alpha(x),$ ⓘ Defines: $\left\langle\NVar{f},\NVar{g}\right\rangle$: inner product over functions Symbols: $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.18.E12 Encodings: TeX, pMML, png See also: Annotations for §1.18(ii), §1.18 and Ch.1

thus generalizing the inner product of (1.18.9). When $\alpha$ is absolutely continuous, i.e. $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x$, see §1.4(v), where the nonnegative weight function $w(x)$ is Lebesgue measurable on $X$. In this section we will only consider the special case $w(x)=1$, so $\,\mathrm{d}\alpha(x)=\,\mathrm{d}x$; in which case $L^{2}\left(X\right)\equiv L^{2}\left(X,\,\mathrm{d}x\right)$.

Assume that $\left\{\phi_{n}\right\}_{n=0}^{\infty}$ is an orthonormal basis of $L^{2}\left(X\right)$. The formulas in §1.18(i) are then:

 1.18.13 $c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,$ $f\in L^{2}\left(X\right)$,
 1.18.14 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|c_{% n}\right|}^{2},$ $f\in L^{2}\left(X\right)$,
 1.18.15 $f(x)=\lim_{m\to\infty}\sum_{n=0}^{m}c_{n}\phi_{n}(x),$ ⓘ Symbols: $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §1.18(ii) Permalink: http://dlmf.nist.gov/1.18.E15 Encodings: TeX, pMML, png See also: Annotations for §1.18(ii), §1.18 and Ch.1

where the limit has to be understood in the sense of $L^{2}$ convergence in the mean:

 1.18.16 $\lim_{m\to\infty}\int_{a}^{b}{\left|f(x)-\sum_{n=0}^{m}c_{n}\phi_{n}(x)\right|% }^{2}\,\mathrm{d}x=0.$

Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_{0}-)+f(x_{0}+))/2$ if $x_{0}$ is an isolated point of discontinuity.

We can rewrite (1.18.15), together with (1.18.13), formally as

 1.18.17 $f(x)=\sum_{n=0}^{\infty}\left\langle f,\phi_{n}\right\rangle\phi_{n}(x)=\int_{% a}^{b}K(x,y)f(y)\,\mathrm{d}y,$

where the integral kernel is given by

 1.18.18 $K(x,y)=\sum_{n=0}^{\infty}\phi_{n}(x)\overline{\phi_{n}(y)}.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $(\NVar{a},\NVar{b})$: open interval and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.18.E18 Encodings: TeX, pMML, png See also: Annotations for §1.18(ii), §1.18 and Ch.1

Thus, in the notation of §1.17, we have an expansion

 1.18.19 $\delta\left(x-y\right)=\sum_{n=0}^{\infty}\phi_{n}(x)\overline{\phi_{n}(y)},$ ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$: complex conjugate and $n$: nonnegative integer Referenced by: §1.18(v), §1.18(vi), §1.18(ii) Permalink: http://dlmf.nist.gov/1.18.E19 Encodings: TeX, pMML, png See also: Annotations for §1.18(ii), §1.18 and Ch.1

of the Dirac delta distribution. Equation (1.18.19) is often called the completeness relation. The analogous orthonormality is

 1.18.20 $\delta_{n,m}=\int_{a}^{b}\phi_{n}(x)\overline{\phi_{m}(x)}\,\mathrm{d}x.$

## §1.18(iii) Linear Operators on a Hilbert Space

### Bounded and Unbounded Linear Operators

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

 1.18.21 $T(\alpha v+\beta w)=\alpha Tv+\beta Tw,$ $v,w\in V$, $\alpha,\beta\in\mathbb{C}$. ⓘ Symbols: $\mathbb{C}$: 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 $\left\|{T}\right\|$ if

 1.18.22 $\left\|{T}\right\|\equiv\sup_{v\in V,\,\left\|{v}\right\|=1}\left\|{Tv}\right% \|<\infty.$

More generally, a linear operator $T$ on $V$ needs not be defined on all of $V$, but only on a linear subspace $\mathcal{D}(T)$ of $V$ which is called the domain of $T$. Then $T\colon\mathcal{D}(T)\to V$ is a linear map. Assume that $\mathcal{D}(T)$ is dense in $V$, i.e., for each $v\in V$ there is a sequence $\{v_{n}\}$ in $\mathcal{D}(T)$ such that $\left\|{v_{n}-v}\right\|\to 0$ as $n\to\infty$. If $\sup_{v\in\mathcal{D}(T),\,\left\|{v}\right\|=1}\left\|{Tv}\right\|$ is finite then $T$ is bounded, and $T$ extends uniquely to a bounded linear operator on $V$. If the supremum is $\infty$, 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 ${T}^{*}$ such that, for $v,w\in V$,

 1.18.23 $\left\langle Tv,w\right\rangle=\left\langle v,{T}^{*}w\right\rangle.$ ⓘ Symbols: ${\NVar{\mathbf{A}}}^{*}$: adjoint of matrix, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: 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 ${T}^{*}=T$, and referred to as symmetric if (1.18.23) holds for $v,w$ in the dense domain $\mathcal{D}(T)$ 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 $Tu_{\lambda}=\lambda u_{\lambda},$ ⓘ 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

$u_{\lambda}\in\mathcal{D}(T)$, corresponding to distinct eigenvalues, are orthogonal: i.e., $\left\langle u_{\lambda},u_{\lambda^{\prime}}\right\rangle=0$, for $\lambda\neq\lambda^{\prime}$. 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 $[a,b]\subset\mathbb{R}$

 1.18.25 $T=\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}},$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 ${T}^{*}$ of $T$ does satisfy $\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle$ where $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x$. We integrate by parts twice giving:

 1.18.26 $\int_{a}^{b}f^{\prime\prime}(x)g(x)\,\mathrm{d}x=\left.f^{\prime}(x)g(x)\right% |^{b}_{a}-\left.f(x)g^{\prime}(x)\right|^{b}_{a}+\int_{a}^{b}f(x)g^{\prime% \prime}(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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 ${T}^{*}=T=\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}},$

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

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

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 $\mathcal{D}(T)=\mathcal{D}({T}^{*})$, 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 $[0,\infty)$, and unbounded operators on them, are the subject of §1.18(ix).

## §1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

Let $X=(a,b)$ be a finite or infinite open interval in $\mathbb{R}$. Consider on $X$ the linear formally self-adjoint second order differential operator

 1.18.28 $\mathcal{L}=-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}+q(x),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ Referenced by: §1.18(viii), §1.18(ix), §1.18(ix), §1.18(v), §1.18(vi), §1.18(iv), §1.18(iv), §1.18(viii), §1.18(viii) Permalink: http://dlmf.nist.gov/1.18.E28 Encodings: TeX, pMML, png See also: Annotations for §1.18(iv), §1.18 and Ch.1

with $q(x)$ real and continuous, unless otherwise noted.

Eigenvalues and eigenfunctions of $T$, self-adjoint extensions of $\mathcal{L}$ with well defined boundary conditions, and utilization of such eigenfunctions for expansion of wide classes of $L^{2}$ functions, will be the focus of the remainder of this section.

The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: $-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}$ being proportional to the kinetic energy operator for a single particle in one dimension, $q(x)$ being proportional to the potential energy, often written as $V(x)$, of that same particle, and which is simply a multiplicative operator. The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. Other applications follow from the fact that $\mathcal{L}$ is suitable for describing vibrations, especially standing waves, which arise in many parts of engineering and the physical sciences, see Birkhoff and Rota (1989, §§10.3 and 10.16). See §18.39(i).

In what follows $T$ will be taken to be a self adjoint extension of $\mathcal{L}$ following the discussion ending the prior sub-section. For $\mathcal{D}(T)$ we can take $C^{2}(X)$, with appropriate boundary conditions, and with compact support if $X$ is bounded, which space is dense in $L^{2}\left(X\right)$, and for $X$ unbounded require that possible non-$L^{2}$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including $\pm\infty$.

Stated informally, the spectrum of $T$ is the set of it’s eigenvalues, these being real as $T$ is self-adjoint. These sets may be discrete, continuous, or a combination of both, as discussed in the following three subsections. Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum.

## §1.18(v) Point Spectra and Eigenfunction Expansions

### General Results

Let $T$ be a self-adjoint extension of differential operator $\mathcal{L}$ of the form (1.18.28) and assume $T$ has a complete set of $L^{2}$ eigenfunctions, $\left\{\phi_{\lambda_{n}}(x)\right\}_{n=0}^{\infty}$ , $x\in X=[a,b]$ this latter being an appropriate sub-set of $\mathbb{R}$, or, in some cases $X=\mathbb{R}$ 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 $\boldsymbol{\sigma}_{p}=\left\{\lambda_{0},\lambda_{1},\dots\right\}$. The eigenfunctions form a complete orthogonal basis in $L^{2}\left(X\right)$, and we can take the basis as orthonormal:

 1.18.29 $\int_{a}^{b}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{m}}(x)}\,\mathrm{d}x% =\delta_{n,m},$

and completeness implies

 1.18.30 $\sum_{n=0}^{\infty}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{n}}(y)}=% \delta\left(x-y\right).$ ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$: 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 $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$, $f(x)$ has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19),

 1.18.31 $f(x)=\sum_{n=0}^{\infty}\phi_{\lambda_{n}}(x)\int_{a}^{b}f(y)\overline{\phi_{% \lambda_{n}}(y)}\,\mathrm{d}y=\sum_{n=0}^{\infty}\widehat{f}(\lambda_{n})\phi_% {\lambda_{n}}(x)$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: 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 $\widehat{f}(\lambda_{n})=\left\langle f,\phi_{\lambda_{n}}\right\rangle.$ ⓘ Symbols: $\left\langle\NVar{f},\NVar{g}\right\rangle$: 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 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|% \widehat{f}(\lambda_{n})\right|}^{2}.$

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 $(Tf)(x)=\sum_{n=0}^{\infty}\lambda_{n}\widehat{f}(\lambda_{n})\phi_{\lambda_{n% }}(x)=\int_{a}^{b}\left(\sum_{n=0}^{\infty}\lambda_{n}\phi_{\lambda_{n}}(x)% \overline{\phi_{\lambda_{n}}(y)}\right)f(y)\,\mathrm{d}y,$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: 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 $(F(T)f)(x)=\int_{a}^{b}\left(\sum_{n=0}^{\infty}F(\lambda_{n})\phi_{\lambda_{n% }}(x)\overline{\phi_{\lambda_{n}}(y)}\right)f(y)\,\mathrm{d}y.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\,\mathrm{d}\NVar{x}$: 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

### Example 1: Three Simple Cases where $q(x)=0$, $X=[0,\pi]$

Possible eigenfunctions of $-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}$ 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: $\phi(0)=\phi(\pi)=0$. The $L^{2}$ normalized eigenfunctions on $[0,\pi]$ are

 1.18.36 $\phi_{\mathrm{sin}}(n,x)=\sqrt{\frac{2}{\pi}}\sin\left(nx\right),$

with $\lambda_{n}=n^{2}$, $n=1,2,3,\dots$.

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

 1.18.37 $\phi_{\mathrm{cos}}(0,x)=\frac{1}{\sqrt{\pi}},\quad\phi_{\mathrm{cos}}(n,x)=% \sqrt{\frac{2}{\pi}}\cos\left(nx\right),$ $n=1,2,3\dots$,

with $\lambda_{n}=n^{2}$, $n=0,1,2,\dots$.

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

 1.18.38 $\phi_{\mathrm{exp}}(\pm n,x)=\frac{1}{\sqrt{\pi}}{\mathrm{e}}^{\pm 2\mathrm{i}% nx},$

with $\lambda_{\pm n}=4n^{2}$, $n=0,1,2,\dots$, 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 $-\infty$ to $\infty$ this multiplicity change is automatically included:

 1.18.39 $\delta\left(x-y\right)=\frac{1}{\pi}\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{2% \mathrm{i}n(x-y)}.$

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

 1.18.40 $f(x)=\frac{1}{\sqrt{\pi}}\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{2\mathrm{i}nx}% \widehat{f}(\lambda_{n}),$

where $\widehat{f}(\lambda_{n})$= $\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}f(y){\mathrm{e}}^{-2\mathrm{i}ny}\,\mathrm{d% }y=\left\langle f,\phi_{\mathrm{exp}}(n)\right\rangle$, 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.

### Hermite’s Differential Equation, $X=(-\infty,\infty)$

The space $X$ is now the full real line, $(-\infty,\infty)$. 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, $n=0,1,2,\ldots$), with eigenvalues $\lambda_{n}=2n+1\in\boldsymbol{\sigma}_{p}$, for the differential operator

 1.18.41 $\mathcal{L}^{\mathrm{Hermite}}=-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}+x^{% 2},$ $x\in X$. ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: 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 $\phi_{n}(x)=\frac{1}{\sqrt{{\pi}^{\frac{1}{2}}2^{n}n!}}{\mathrm{e}}^{-x^{2}/2}% H_{n}\left(x\right),$

(1.18.31) becomes

 1.18.43 $f(x)=\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}\frac{{\mathrm{e}}^{-(x^{2}+y^{% 2})/2}}{{\pi}^{\frac{1}{2}}2^{n}n!}H_{n}\left(x\right)H_{n}\left(y\right)f(y)% \,\mathrm{d}y,$

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 $\phi_{n}(x)$ and $\phi_{n}^{\prime}(x)$ vanish as $x\to\pm\infty$, which in this case is equivalent to requiring $\phi_{n}(x)\in L^{2}\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point.

## §1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

### General Results

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 $\mathcal{L}$ of the form (1.18.28) on an unbounded interval $X\subset\mathbb{R}$, which we will take as $X=[0,+\infty)$, and assume that $q(x)\to 0$ monotonically as $x\to\infty$, 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 $\boldsymbol{\sigma}$ of $T$ consists only of a continuous spectrum, referred to as $\boldsymbol{\sigma}_{c}$. In this subsection it is assumed that $\boldsymbol{\sigma}_{c}=[0,\infty)$. 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 $\int_{0}^{\infty}{\phi_{\lambda}}(x)\overline{\phi_{\lambda^{\prime}}(x)}\,% \mathrm{d}x=\delta\left(\lambda-\lambda^{\prime}\right),$ $\lambda,\lambda^{\prime}\in\boldsymbol{\sigma}_{c}$,

and completeness relation

 1.18.45 $\delta\left(x-y\right)=\int_{0}^{\infty}\phi_{\lambda}(x)\overline{\phi_{% \lambda}(y)}\,\mathrm{d}\lambda.$

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 $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$, $f(x)$ has the eigenfunction expansion, analogous to that of (1.18.33),

 1.18.46 $f(x)=\int_{0}^{\infty}\phi_{\lambda}(x)\widehat{f}(\lambda)\,\mathrm{d}\lambda,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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 $\widehat{f}(\lambda)=\left\langle f,\phi_{\lambda}\right\rangle.$ ⓘ Symbols: $\left\langle\NVar{f},\NVar{g}\right\rangle$: 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 $\int_{0}^{\infty}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\int_{0}^{\infty}{\left|% \widehat{f}(\lambda)\right|}^{2}\,\mathrm{d}\lambda.$

The analog of (1.18.34) is

 1.18.49 $(Tf)(x)=\int_{0}^{\infty}\left(\int_{0}^{\infty}\lambda\phi_{\lambda}(x)% \overline{\phi_{\lambda}(y)}\,\mathrm{d}\lambda\right)f(y)\,\mathrm{d}y,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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 $(F(T)f)(x)=\int_{0}^{\infty}\left(\int_{0}^{\infty}F(\lambda)\phi_{\lambda}(x)% \overline{\phi_{\lambda}(y)}\,\mathrm{d}\lambda\right)f(y)\,\mathrm{d}y.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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 $\left\langle F(T)f,f\right\rangle=\int_{0}^{\infty}F(\lambda){\left|\widehat{f% }(\lambda)\right|}^{2}\,\mathrm{d}\lambda.$

In particular, this holds for $F(\lambda)=\left(z-\lambda\right)^{-1}$, $z\in\mathbb{C}\setminus\boldsymbol{\sigma}_{c}$

 1.18.52 $\left\langle\left(z-T\right)^{-1}f,f\right\rangle=\int_{\boldsymbol{\sigma}}{% \left|\widehat{f}(\lambda)\right|}^{2}\frac{\,\mathrm{d}\lambda}{z-\lambda},$ $f\in L^{2}\left(X\right)$, $z\notin\boldsymbol{\sigma}_{c}$,

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 $\lim_{\nu\to 0+}\frac{1}{2\pi\mathrm{i}}\left(\left\langle\left(\mu-\mathrm{i}% \nu-T\right)^{-1}f,f\right\rangle-\left\langle\left(\mu+\mathrm{i}\nu-T\right)% ^{-1}f,f\right\rangle\right)$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\left\langle\NVar{f},\NVar{g}\right\rangle$: 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 $\mu\in\boldsymbol{\sigma}_{c}$, and $0$ otherwise. This is the discontinuity across the branch cut in (1.18.52) $\boldsymbol{\sigma}_{c}\subset\mathbb{R}$, from $z$ below to above the cut, divided by $2\pi\mathrm{i}$. See Newton (2002, §§7.1 and 7.3).

More generally, for $f\in C(X)$, $x\in X$, see (1.4.24),

 1.18.54 $\lim_{\epsilon\to 0{+}}\int_{X}\frac{f(y)}{x\pm\mathrm{i}\epsilon-y}\,\mathrm{% d}y=P\pvint_{X}\frac{f(y)}{x-y}\,\mathrm{d}y\mp\mathrm{i}\pi f(x).$

### Example 1: Bessel–Hankel Transform, $X=[0,\infty)$

By Bessel’s differential equation in the form (10.13.1) we have the functions $\sqrt{x}J_{\nu}\left(x\sqrt{\lambda}\right)$ ($\lambda\geq 0$, for $J_{\nu}$ see §10.2(ii)) as eigenfunctions with eigenvalue $\lambda$ of the self-adjoint extension of the differential operator

 1.18.55 $\mathcal{L}^{\mathrm{Bessel}}=-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}+% \frac{\nu^{2}-\frac{1}{4}}{x^{2}},$ $x\in X$. ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $\in$: element of Permalink: http://dlmf.nist.gov/1.18.E55 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta\left(x-y\right)=\frac{1}{\sqrt{xy}}\delta\left(x-y\right)$,

 1.18.56 $\delta\left(x-y\right)=\int_{0}^{\infty}\sqrt{xt}J_{\nu}\left(xt\right)\sqrt{% yt}J_{\nu}\left(yt\right)\,\mathrm{d}t,$ $\Re\nu>-1$, $x,y\geq 0$.

For $f(x)$ piecewise continuously differentiable on $[0,\infty)$

 1.18.57 $\lim_{R\to\infty}\int_{0}^{\infty}f(y)\left(\int_{0}^{R}\sqrt{xt}J_{\nu}\left(% xt\right)\sqrt{yt}J_{\nu}\left(yt\right)\,\mathrm{d}t\right)\,\mathrm{d}y=% \tfrac{1}{2}\left(f(x+)+f(x-)\right),$ $x>0$, $\nu>-1$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Proved: Li and Wong (2008, §3, Theorem 1)(proved) Referenced by: §1.18(vi), §1.18(vi) Permalink: http://dlmf.nist.gov/1.18.E57 Encodings: TeX, pMML, png See also: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

provided that:

 1.18.58 $\displaystyle\int_{1}^{\infty}y^{-1}\left|f(y)\right|\,\mathrm{d}y$ $\displaystyle<\infty,$ $\displaystyle\int_{0}^{1}\left(1+y^{\nu+\frac{1}{2}}\right)\left|f(y)\right|\,% \mathrm{d}y$ $\displaystyle<\infty.$ ⓘ

(1.18.57) is the Hankel transform (10.22.76)–(10.22.77). See Titchmarsh (1962a, pp. 87–90) for a first principles derivation for the case $\Re\nu\geq 1$.

For generalizations see the Weber transform (10.22.78) and an extended Bessel transform (10.22.79).

### Example 2: Sine and Cosine Transforms, $X=[0,\infty)$

The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for $\nu=\pm\frac{1}{2}$ the Bessel functions reduce to the trigonometric functions, see (10.16.1).

More generally,

 1.18.59 $f(x)=\frac{1}{\pi}\int_{-\infty}^{\infty}\left(\int_{0}^{\infty}\cos\left(xt% \right)\cos\left(yt\right)\,\mathrm{d}t\right)f(y)\,\mathrm{d}y+\frac{1}{\pi}% \int_{-\infty}^{\infty}\left(\int_{0}^{\infty}\sin\left(xt\right)\sin\left(yt% \right)\,\mathrm{d}t\right)f(y)\,\mathrm{d}y,$ $x\in\mathbb{R}$.

For $f(x)$ even in $x$ this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for $f(x)$ odd the Fourier sine transform pair (1.14.10) & (1.14.12). These latter results also correspond to use of the $\delta\left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2).

## §1.18(vii) Continuous Spectra: More General Cases

More generally, continuous spectra may occur in sets of disjoint finite intervals $[\lambda_{a},\lambda_{b}]\in(0,\infty)$, often called bands, when $q(x)$ is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). Should $q(x)$ be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). For example, replacing $2q\cos{(2z)}$ of (28.2.1) by $\lambda\cos{(2\pi\alpha n+\theta)}$, $n\in\mathbb{Z}$ gives an almost Mathieu equation which for appropriate $\alpha$ has such properties.

## §1.18(viii) Mixed Spectra and Eigenfunction Expansions

In general, operators $T$ being formally self-adjoint second order differential operators of the form (1.18.28), with $X$ unbounded, will have both a continuous and a point spectrum, and thus, correspondingly, $non-L^{2}\left(X\right)$ eigenfunctions as in §1.18(vi) and $L^{2}\left(X\right)$ eigenfunctions as in §1.18(v). We assume a continuous spectrum $\lambda\in\boldsymbol{\sigma}_{c}=[0,\infty)$, and a finite or countably infinite point spectrum $\boldsymbol{\sigma}_{p}$ with elements $\lambda_{n}$. In what follows, integrals over the continuous parts of the spectrum will be denoted by $\boldsymbol{\sigma}_{c}$, and sums over the discrete spectrum by $\boldsymbol{\sigma}_{p}$, with $\boldsymbol{\sigma}=\boldsymbol{\sigma}_{c}\cup\boldsymbol{\sigma}_{p}$ denoting the full spectrum. It is to be noted that if any of the $\lambda\in\boldsymbol{\sigma}$ have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. Then orthogonality and normalization relations are

 1.18.60 $\int_{X}\phi_{\lambda}(x)\overline{\phi_{\lambda^{\prime}}(x)}\,\mathrm{d}x=% \delta\left(\lambda-\lambda^{\prime}\right),$ $\lambda,\lambda^{\prime}\in\boldsymbol{\sigma}_{c}$,
 1.18.61 $\int_{X}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{m}}(x)}\,\mathrm{d}x=% \delta_{n,m},$ $\lambda_{n},\lambda_{m}\in\boldsymbol{\sigma}_{p}$,
 1.18.62 $\int_{X}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda}(x)}\,\mathrm{d}x=0,$ $\lambda_{n}\in\boldsymbol{\sigma}_{p}$, $\lambda\in\boldsymbol{\sigma}_{c}$,

compare (1.18.29) and (1.18.44). The formal completeness relation is now

 1.18.63 $\delta\left(x-x^{\prime}\right)=\sum_{\boldsymbol{\sigma}_{p}}\phi_{\lambda_{n% }}(x)\overline{\phi_{\lambda_{n}}(x^{\prime})}+\int_{\boldsymbol{\sigma}_{c}}% \phi_{\lambda}(x)\overline{\phi_{\lambda}(x^{\prime})}\,\mathrm{d}\lambda,$ $x,x^{\prime}\in X$,

compare (1.18.30) and (1.18.45), and the eigenfunction expansions are of the form

 1.18.64 $f(x)=\int_{\boldsymbol{\sigma}_{c}}\widehat{f}(\lambda)\phi_{\lambda}(x)\,% \mathrm{d}\lambda+\sum_{\boldsymbol{\sigma}_{p}}\widehat{f}(\lambda_{n})\phi_{% \lambda_{n}}(x),$ $f(x)\in C(X)\cap L^{2}\left(X\right)$.

Note that the notations of (1.18.32) and (1.18.47) are used to distinguish the contributions from the discrete and continuous parts of the spectrum. Then

 1.18.65 $\int_{X}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\int_{\boldsymbol{\sigma}_{c}}{% \left|\widehat{f}(\lambda)\right|}^{2}\,\mathrm{d}\lambda+\sum_{\boldsymbol{% \sigma}_{p}}{\left|\widehat{f}(\lambda_{n})\right|}^{2},$ $f\in L^{2}\left(X\right)$.

The analogs of (1.18.49)–(1.18.52) may be written in a similar fashion each now including contributions from both the discrete and continuous parts of the spectrum, as in (1.18.65). Showing one, representative, example: the analog of (1.18.52) is now

 1.18.66 $\left\langle\left(z-T\right)^{-1}f,f\right\rangle=\sum_{\boldsymbol{\sigma}_{p% }}\frac{{\left|\widehat{f}(\lambda_{n})\right|}^{2}}{z-\lambda_{n}}+\int_{% \boldsymbol{\sigma}_{c}}{\left|\widehat{f}(\lambda)\right|}^{2}\frac{\,\mathrm% {d}\lambda}{z-\lambda},$ $f\in L^{2}\left(X\right)$, $z\notin\boldsymbol{\sigma}$.

This representation has poles with residues ${\left|\widehat{f}(\lambda_{n})\right|}^{2}$ at the discrete eigenvalues and a branch cut along $[0,\infty)$ with discontinuity, from below to above the cut, $2\pi\mathrm{i}{\left|\widehat{f}(\lambda)\right|}^{2}$, as in (1.18.53), see Newton (2002, §7.1.1).

Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies $\lambda_{\mathrm{res}}-\mathrm{i}\Gamma_{\mathrm{res}}/2$ corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to $1/\Gamma_{\mathrm{res}}$. For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change $x\to x{\mathrm{e}}^{\mathrm{i}\theta}$, in $\mathcal{L}$, which rotates the continuous spectrum $\boldsymbol{\sigma}_{c}\to\boldsymbol{\sigma}_{c}{\mathrm{e}}^{-2\mathrm{i}\theta}$ and the branch cut of (1.18.66) into the lower half complex plain by the angle $-2\theta$, with respect to the unmoved branch point at $\lambda=0$; thus, providing access to resonances on the higher Riemann sheet should $\theta$ be large enough to expose them. This dilatation transformation, which does require analyticity of $q(x)$ in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $\left\langle\left(z-T\right)^{-1}f,f\right\rangle$.

### Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

Suppose that $X$ is the whole real line in one dimension, and that $q(x)$, in (1.18.28) has (non-oscillatory) limits of $0$ at both $\pm\infty$, and thus a continuous spectrum on $\boldsymbol{\sigma}\geq 0$. What then is the condition on $q(x)$ to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if $q(x)\equiv 0$ then there is only a continuous spectrum. Surprisingly, if $q(x)<0$ on any interval on the real line, even if positive elsewhere, as long as $\int_{X}q(x)\,\mathrm{d}x\leq 0$, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding $L^{2}\left(X\right)$ eigenfunction. Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha\delta\left(x-a\right)$, $\alpha>0$, there will be an $L^{2}$ eigenfunction localized in the vicinity of $x=a$, with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\infty)$. Similar results hold for two, but not higher, dimensional quantum systems. See Brownstein (2000) and Yang and de Llano (1989) for numerical examples, based on variational calculations, and Simon (1976) and Chadan et al. (2003) for rigorous mathematical discussion.

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

Consider formally self-adjoint operators of the form

 1.18.67 $\mathcal{L}_{\ell}=-\frac{1}{2}\frac{{\mathrm{d}}^{2}}{{\mathrm{d}r}^{2}}+% \frac{\ell(\ell+1)}{2r^{2}}+V(r),$ $\ell=0,1,2,\dots$, $V(r)\to 0$ as $r\to\infty$, ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $\ell$: integer Permalink: http://dlmf.nist.gov/1.18.E67 Encodings: TeX, pMML, png See also: Annotations for §1.18(viii), §1.18(viii), §1.18 and Ch.1

which appear in the quantum theory of binding or scattering of a particle in a spherically symmetric potential $V(r)$ in three dimensions, and where $r\in[0,\infty)$. The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, $\boldsymbol{\sigma}_{c}=[0,\infty)$, or, said more simply, in the continuum. See §18.39 for discussion of Schrödinger equations and operators. For fixed angular momentum $\ell$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues $\lambda_{n},n=0,1,\dots,N-1$, with corresponding $L^{2}\left([0,\infty),r^{2}\,\mathrm{d}r\right)$ eigenfunctions $\phi_{n}(r)$, and also a continuous spectrum $\lambda\in[0,\infty)$, with Dirac-delta normalized eigenfunctions $\phi_{\lambda}(r)$, also with measure $r^{2}\,\mathrm{d}r$. Unlike in the example in the paragraph above, in 3-dimensions a “dip below zero, or a potential well” in $V(r)$ does not always correspond to the existence of a discrete part of the spectrum. The well must be deep and broad enough to allow existence of such $L^{2}$ discrete states. The number, $N$, of discrete states depends on the nature of $V(r)$, as well as $\ell$, and, again, $V(r)$ must vanish as $r\to\infty$, corresponding to the traditionally assumed start of the energy continuum at $\lambda=0$. In unusual cases $N=\infty$, even for all $\ell$, 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). See Bethe and Salpeter (1977, Ch. 1, (4.12)–(4.13)) for the resulting transform pair in this case.

## §1.18(ix) Mathematical Background

If $T$ is an unbounded linear operator on a Hilbert space $V$ with dense domain $\mathcal{D}(T)$ then the adjoint ${T}^{*}$ of $T$ is the linear operator with domain

 1.18.68 $\mathcal{D}({T}^{*})=\left\{w\in V\,\Big{|}\,\sup_{v\in\mathcal{D}(T),\,\left% \|{v}\right\|=1}\left|\left\langle Tv,w\right\rangle\right|<\infty\right\},$

such that

 1.18.69 $\left\langle Tv,w\right\rangle=\left\langle v,{T}^{*}w\right\rangle,$ $v\in\mathcal{D}(T)$, $w\in\mathcal{D}({T}^{*})$.

A linear operator $T$ with dense domain is called symmetric if

 1.18.70 $\left\langle Tv,w\right\rangle=\left\langle v,Tw\right\rangle,$ $v,w\in\mathcal{D}(T)$. ⓘ Symbols: $\in$: element of, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: inner product over vectors and $w$: variable Permalink: http://dlmf.nist.gov/1.18.E70 Encodings: TeX, pMML, png See also: Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

If $T$ is symmetric then $T\subset{T}^{*}$, i.e., $\mathcal{D}(T)\subset\mathcal{D}({T}^{*})$ and ${T}^{*}v=Tv$ for $v\in\mathcal{D}(T)$. Then also $T\subset T^{**}\subset{T}^{*}$, where $T^{**}={({T}^{*})}^{*}$. If $T\subset T^{**}={T}^{*}$ then $T$ is essentially self-adjoint and if $T={T}^{*}$ then $T$ is self-adjoint.

### Spectrum of an Operator

Let $T$ be a linear operator on $V$ with dense domain $\mathcal{D}(T)$ and with range $\mathcal{R}(T)=\{Tv\mid v\in\mathcal{D}(T)\}$. Such an operator $T$ is called injective if, for any $u,v$ in its domain, $Tu=Tv$ implies that $u=v$. The resolvent set $\rho(T)$ consists of all $z\in\mathbb{C}$ such that (i) $z-T$ is injective, (ii) $\mathcal{R}(z-T)$ is dense in $V$, (iii) the resolvent $\left(z-T\right)^{-1}$ is bounded. The spectrum $\boldsymbol{\sigma}(T)$ is the complement in $\mathbb{C}$ of $\rho(T)$. The spectrum $\boldsymbol{\sigma}(T)$ is the disjoint union of three sets:

1. 1.

The point spectrum $\boldsymbol{\sigma}_{p}$. It consists of all $z\in\mathbb{C}$ for which $z-T$ is not injective, or equivalently, for which $z$ is an eigenvalue of $T$, i.e., $Tv=zv$ for some $v\in\mathcal{D}(T)\backslash\{0\}$.

2. 2.

The continuous spectrum $\boldsymbol{\sigma}_{c}$. It consists of all $z\in\mathbb{C}$ for which $z-T$ is injective and has dense range, but $\left(T-z\right)^{-1}$ is not bounded.

3. 3.

The residual spectrum. It consists of all $z\in\mathbb{C}$ for which $z-T$ is injective, but does not have dense range.

If $T$ is a bounded operator then its spectrum is a closed bounded subset of $\mathbb{C}$. If $T$ is self-adjoint (bounded or unbounded) then $\sigma(T)$ is a closed subset of $\mathbb{R}$ and the residual spectrum is empty. Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. If an eigenvalue is of multiplicity greater than $1$ then an orthonormal basis of eigenfunctions can be given for the eigenspace.

### Self-adjoint extensions of a symmetric Operator

Let $T$ be a symmetric operator on a Hilbert space $V$, so $\mathcal{D}(T)$ is dense in $V$ and $T\subset T^{**}\subset{T}^{*}$. For $z\in\mathbb{C}\backslash\mathbb{R}$ let $N_{z}$ be the $z$-eigenspace of ${T}^{*}$, i.e., $N_{z}$ is the linear subspace of $\mathcal{D}({T}^{*})$ consisting of all $v$ for which ${T}^{*}v=zv$. Then $\dim N_{z}$ is constant for $\Im z>0$ and also constant for $\Im z<0$. Put $n_{+}=\dim N_{z}$ ($\Im z>0$) and $n_{-}=\dim N_{z}$ ($\Im z<0$), the deficiency indices for $T$. Then $T$ has self-adjoint extensions iff $n_{+}=n_{-}$. We have a direct sum of linear spaces: $\mathcal{D}({T}^{*})=\mathcal{D}(T^{**})+N_{\mathrm{i}}+N_{-\mathrm{i}}$. Assume $n_{+}=n_{-}$. Then any self-adjoint extension of $T$ is determined by a linear isometry $U\colon N_{\mathrm{i}}\to N_{-\mathrm{i}}$ and it is the restriction of ${T}^{*}$ to $\{v+w+Uw\mid v\in\mathcal{D}(T^{**}),\;w\in N_{\mathrm{i}}\}$.

### Self-adjoint extensions of (1.18.28) and the Weyl alternative

For a formally self-adjoint second order differential operator $\mathcal{L}$, such as that of (1.18.28), the space $\mathcal{D}({\mathcal{L}}^{*})$ can be seen to consist of all $f\in L^{2}\left(X\right)$ such that the distribution $\mathcal{L}f$ can be identified with a function in $L^{2}\left(X\right)$, which is the function ${\mathcal{L}}^{*}f$. Then, for $z\in\mathbb{C}\backslash\mathbb{R}$, $f\in N_{z}$ iff $f$ is an ordinary solution (i.e., $f\in C^{2}(X)$) of $Lf=zf$ which is moreover in $L^{2}\left(X\right)$. Thus $N_{z}$ has dimension 0, 1 or 2. Also, because $q$ is real-valued, $f\in N_{z}$ iff $\overline{f}\in N_{\overline{z}}$. So $\mathcal{L}$ has self-adjoint extensions with deficiency indices $n_{+}=n_{-}=0$, or $1$ or $2$. Pick $c\in(a,b)$. Let $n_{1},n_{1}$ be the deficiency indices for $\mathcal{L}$ restricted to $(a,c)$, and $n_{2},n_{2}$ the ones for $\mathcal{L}$ restricted to $(c,b)$. Then $n_{1}$ and $n_{2}$ are independent of $c$. By Weyl’s alternative $n_{1}$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_{2}$. The two (equal) deficiency indices of $\mathcal{L}$ are then equal to $n_{1}+n_{2}-2$. A boundary value for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{*})$ of the form

 1.18.71 $\mathcal{B}(f)=\lim_{x\to a+}\left(\alpha(x)f(x)+\beta(x)f^{\prime}(x)\right),$ $f\in\mathcal{D}({\mathcal{L}}^{*})$, ⓘ Symbols: ${\NVar{\mathbf{A}}}^{*}$: adjoint of matrix and $\in$: element of Referenced by: Erratum (V1.2.0) Section 1.18 Permalink: http://dlmf.nist.gov/1.18.E71 Encodings: TeX, pMML, png See also: Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

where $\alpha$ and $\beta$ are given functions on $X$, and where the limit has to exist for all $f$. Then, if the linear form $\mathcal{B}$ is nonzero, the condition $\mathcal{B}(f)=0$ is called a boundary condition at $a$. Boundary values and boundary conditions for the end point $b$ are defined in a similar way. If $n_{1}=1$ then there are no nonzero boundary values at $a$; if $n_{1}=2$ then the above boundary values at $a$ form a two-dimensional class. Similarly at $b$. Any self-adjoint extension of $\mathcal{L}$ can be obtained by restricting ${\mathcal{L}}^{*}$ to those $f\in\mathcal{D}({\mathcal{L}}^{*})$ for which, if $n_{1}=2$, $\mathcal{B}_{1}(f)=0$ for a chosen $\mathcal{B}_{1}$ at $a$ and, if $n_{2}=2$, $\mathcal{B}_{2}(f)=0$ for a chosen $\mathcal{B}_{2}$ at $b$.

Integral transforms (10.22.78) and (10.22.79) are examples of the utility of these extensions.

### Spectral expansions and self-adjoint extensions

The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results.

## §1.18(x) Literature

The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. In parallel, similar, and more general formulations have grown out of functional analysis itself, as in the work of Stone (1990), Rudin (1973), Reed and Simon (1980), Reed and Simon (1975), Reed and Simon (1978), Reed and Simon (1979), Cycon et al. (2008), Dunford and Schwartz (1988, Ch. XIII), Hall (2013, pp. 127-223). Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.