1 Algebraic and Analytic Methods Topics of Discussion 1.17 Integral and Series Representations of the Dirac Delta 2 Asymptotic Approximations

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Keywords:: differential operators, eigenfunction expansions, linear, second order
Referenced by:: §1.18(iii), §18.39(i), §18.39(i), Erratum (V1.2.0) §1.18, Erratum (V1.2.0) §1.18, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.18
Addition (effective with 1.2.0):: This section was added.
See also:: Annotations for Ch.1

§1.18(i) Hilbert spaces
§1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$
§1.18(iii) Linear Operators on a Hilbert Space
§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators
§1.18(v) Point Spectra and Eigenfunction Expansions
§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases
§1.18(vii) Continuous Spectra: More General Cases
§1.18(viii) Mixed Spectra and Eigenfunction Expansions
§1.18(ix) Mathematical Background
§1.18(x) Literature

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

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Keywords:: Hilbert, Hilbert space, expansion of arbitrary vector, inner product, $\ell^{2}$ space, space
Referenced by:: §1.18(ii), §1.18, Erratum (V1.2.0) §1.18, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.18.i
See also:: Annotations for §1.18 and Ch.1

A complex linear vector space 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 ; (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 . With norm defined by

1.18.1 $\left\|{v}\right\|=\sqrt{\left\langle v,v\right\rangle},$

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

becomes a normed linear vector space. If $\left\|{v}\right\|=1$ then is normalized. Two elements and in 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 is called a Hilbert space if every Cauchy sequence $\{v_{n}\}$ in (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 Bessel’s inequality holds:

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

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Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E2
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See also:: Annotations for §1.18(i), §1.18 and Ch.1

where $v\in V$ and

1.18.3 $c_{n}=\left\langle v,v_{n}\right\rangle.$

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Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and : 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 is separable if there is an (at most countably infinite) orthonormal set $\{v_{n}\}$ in such that for every $v\in V$

1.18.4 $\sum_{n}{\left|c_{n}\right|}^{2}={\left\|{v}\right\|}^{2},$

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Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), : 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 $\sum_{n=0}^{\infty}{\left|c_{n}\right|}^{2}<\infty.$

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Symbols:: : 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 can be given by

1.18.6 $v=\sum_{n=0}^{\infty}c_{n}v_{n},$

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Symbols:: : 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 $\lim_{N\to\infty}\left\|{v-\sum_{n=0}^{N}c_{n}v_{n}}\right\|=0.$

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Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²) and : 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 ${\left\|{v}\right\|}^{2}=\sum_{n=0}^{\infty}{\left|c_{n}\right|}^{2}<\infty.$

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Symbols:: $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(i), §1.18 and Ch.1

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

1.18.9 $\left\langle v,w\right\rangle=\sum_{n=0}^{\infty}c_{n}\overline{d_{n}}.$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, : variable and : 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 can be made isomorphic to $\ell^{2}$ by choosing a complete orthonormal set $\left\{v_{n}\right\}_{n=0}^{\infty}$ in . Then an isomorphism is given by

1.18.10 $\sum_{n=0}^{\infty}c_{n}v_{n}\mapsto(c_{0},c_{1},c_{2},\ldots)\colon V\to\ell^% {2}.$

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Symbols:: $\ell$ : integer and : 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 $\mathbb{R}$

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Defines:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions
Keywords:: Hilbert space, $L^{2}$ , Stieltjes measure, completeness relation, completness relation, discrete spectrum, eigenfunctions, expansion of arbitrary function, inner product, ortho-normality relation
Referenced by:: §1.18(v)
Permalink:: http://dlmf.nist.gov/1.18.ii
See also:: Annotations for §1.18 and Ch.1

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

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

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

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 $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)\overline{g(x)}\,\mathrm{d}% \alpha(x),$

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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 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 is Lebesgue measurable on . In this section we will only consider the special case , 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 $c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,$ $f\in L^{2}\left(X\right)$ ,

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Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral and : nonnegative integer
Referenced by:: §1.18(v), §1.18(vi), §1.18(ii)
Permalink:: http://dlmf.nist.gov/1.18.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

1.18.14 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|c_{% n}\right|}^{2},$ $f\in L^{2}\left(X\right)$ ,

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Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

1.18.15 $f(x)=\lim_{m\to\infty}\sum_{n=0}^{m}c_{n}\phi_{n}(x),$

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

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where 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 $f(x)=\sum_{n=0}^{\infty}\left\langle f,\phi_{n}\right\rangle\phi_{n}(x)=\int_{% a}^{b}K(x,y)f(y)\,\mathrm{d}y,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval and : nonnegative integer
Referenced by:: §1.18(v)
Permalink:: http://dlmf.nist.gov/1.18.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

where the integral kernel is given by

1.18.18 $K(x,y)=\sum_{n=0}^{\infty}\phi_{n}(x)\overline{\phi_{n}(y)}.$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $(\NVar{a},\NVar{b})$ : open interval and : 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 $\delta\left(x-y\right)=\sum_{n=0}^{\infty}\phi_{n}(x)\overline{\phi_{n}(y)},$

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Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate and : 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 $\delta_{n,m}=\int_{a}^{b}\phi_{n}(x)\overline{\phi_{m}(x)}\,\mathrm{d}x.$

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer and : nonnegative integer
Referenced by:: §1.18(v), §1.18(vi), §1.18(vi)
Permalink:: http://dlmf.nist.gov/1.18.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ii), §1.18 and Ch.1

§1.18(iii) Linear Operators on a Hilbert Space

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Notes:: See https://www.physicsoverflow.org/27488/physical-interpretation-application-residual-spectrum-operator for a sketch of proof that the residual spectrum of a selfadjoint operator is empty.
Permalink:: http://dlmf.nist.gov/1.18.iii
See also:: Annotations for §1.18 and Ch.1

Bounded and Unbounded Linear Operators

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Keywords:: Hilbert space, linear operators
See also:: Annotations for §1.18(iii), §1.18 and Ch.1

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

1.18.21 $T(\alpha v+\beta w)=\alpha Tv+\beta Tw,$ $v,w\in V$ , $\alpha,\beta\in\mathbb{C}$ .

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Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of and : 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 be a Hilbert space. A linear operator on is bounded with norm $\left\|{T}\right\|$ if

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

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Symbols:: $\in$ : element of, $\equiv$ : equals by definition, $\left\|{\NVar{\mathbf{A}}}\right\|$ : matrix norm, $\sup$ : least upper bound (supremum) and $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (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 on needs not be defined on all of , but only on a linear subspace $\mathcal{D}(T)$ of which is called the domain of . Then $T\colon\mathcal{D}(T)\to V$ is a linear map. Assume that $\mathcal{D}(T)$ is dense in , 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 is bounded, and extends uniquely to a bounded linear operator on . If the supremum is $\infty$ , then is an unbounded linear operator on .

Self-Adjoint and Symmetric Operators

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Keywords:: Hilbert space, formally self-adjoint operators, symmetric operators
See also:: Annotations for §1.18(iii), §1.18 and Ch.1

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

1.18.23 $\left\langle Tv,w\right\rangle=\left\langle v,{T}^{*}w\right\rangle.$

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

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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 , the eigenfunctions may always be orthogonalized in this degenerate sub-space.

Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions

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Keywords:: Hilbert space, formally self-adjoint differential operators, self-adjoint differential operators, self-adjoint extensions of differential operators
See also:: Annotations for §1.18(iii), §1.18 and Ch.1

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 in the finite interval $[a,b]\subset\mathbb{R}$

1.18.25 $T=\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}},$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to
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 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 $\int_{a}^{b}f^{\prime\prime}(x)g(x)\,\mathrm{d}x=\left.f^{\prime}(x)g(x)\right% |^{b}_{a}-\left.f(x)g^{\prime}(x)\right|^{b}_{a}+\int_{a}^{b}f(x)g^{\prime% \prime}(x)\,\mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of 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 ${T}^{*}=T=\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}},$

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Symbols:: ${\NVar{\mathbf{A}}}^{*}$ : adjoint of matrix and $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to
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 is said to be formally self adjoint.

For 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 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 and 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 , and , 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 and satisfy the same boundary conditions and thus define the same domains. Thus is indeed self adjoint.

Other choices of boundary conditions, identical for and , and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of . 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

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Keywords:: Liouville normal form, Schrödinger operator, eigenvalue, formally self adjoint linear operator, linear second order differential operator, multiplicity, set of eigenvalues, taking multiplicities into account, spectrum of a self-adjoint extension of a linear differential operator
Permalink:: http://dlmf.nist.gov/1.18.iv
See also:: Annotations for §1.18 and Ch.1

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

1.18.28 $\mathcal{L}=-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}x}^{2}}+q(x),$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to
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 real and continuous, unless otherwise noted.

Eigenvalues and eigenfunctions of , 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, being proportional to the potential energy, often written as , 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 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 is bounded, which space is dense in $L^{2}\left(X\right)$ , and for 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 is the set of it’s eigenvalues, these being real as 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

ⓘ

Keywords:: discrete spectra, eigenfunction expansions, second order differential operators
Referenced by:: §1.17(ii), §1.18(viii), §18.39(i), §18.39(i)
Permalink:: http://dlmf.nist.gov/1.18.v
See also:: Annotations for §1.18 and Ch.1

General Results

ⓘ

Keywords:: completeness relation, completness relation, discrete spectrum, eigenfunctions
See also:: Annotations for §1.18(v), §1.18 and Ch.1

Let be a self-adjoint extension of differential operator $\mathcal{L}$ of the form (1.18.28) and assume 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 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 $\int_{a}^{b}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{m}}(x)}\,\mathrm{d}x% =\delta_{n,m},$

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer and : 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 $\sum_{n=0}^{\infty}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{n}}(y)}=% \delta\left(x-y\right).$

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate and : 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 $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$ , 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 , $\int$ : integral and : 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 : 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 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|% \widehat{f}(\lambda_{n})\right|}^{2}.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 , and of functions of , these being expansions of and 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 , $\int$ : integral and : 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 , $\int$ : integral and : 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 , $X=[0,\pi]$

ⓘ

Keywords:: Fourier eigenfunction expansion, discrete spectrum
See also:: Annotations for §1.18(v), §1.18 and Ch.1

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $(\NVar{a},\NVar{b})$ : open interval, $\sin\NVar{z}$ : sine function and : 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 $\lambda_{n}=n^{2}$ , $n=1,2,3,\dots$ .

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

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $(\NVar{a},\NVar{b})$ : open interval and : 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 $\lambda_{n}=n^{2}$ , $n=0,1,2,\dots$ .

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $(\NVar{a},\NVar{b})$ : open interval and : 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 $\lambda_{\pm n}=4n^{2}$ , $n=0,1,2,\dots$ , with all eigenvalues, for $\left|n\right|>0$ , having multiplicity two, as changing the sign of changes the eigenfunction but not the eigenvalue, and multiplicity one for . Letting run from $-\infty$ to $\infty$ this multiplicity change is automatically included:

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

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : 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 : 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 $f(x)=\frac{1}{\sqrt{\pi}}\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{2\mathrm{i}nx}% \widehat{f}(\lambda_{n}),$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : 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 $\widehat{f}(\lambda_{n})$ = $\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}f(y){\mathrm{e}}^{-2\mathrm{i}ny}\,\mathrm{d% }y=\left\langle f,\phi_{\mathrm{exp}}(n)\right\rangle$ , 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, $X=(-\infty,\infty)$

ⓘ

Keywords:: Hermite differential operator, eigenfunction expansion
See also:: Annotations for §1.18(v), §1.18 and Ch.1

The space 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 with respect to 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 $\phi_{n}(x)=\frac{1}{\sqrt{{\pi}^{\frac{1}{2}}2^{n}n!}}{\mathrm{e}}^{-x^{2}/2}% H_{n}\left(x\right),$

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ) and : 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 $f(x)=\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}\frac{{\mathrm{e}}^{-(x^{2}+y^{% 2})/2}}{{\pi}^{\frac{1}{2}}2^{n}n!}H_{n}\left(x\right)H_{n}\left(y\right)f(y)% \,\mathrm{d}y,$

ⓘ

Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\int$ : integral and : 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 $\phi_{n}(x)$ 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

ⓘ

Keywords:: continuous spectra, eigenfunction expansions, second order differential operators
Referenced by:: §1.17(ii), §1.18(viii), §1.18(viii)
Permalink:: http://dlmf.nist.gov/1.18.vi
See also:: Annotations for §1.18 and Ch.1

General Results

ⓘ

Keywords:: completeness relation, continuous spectrum, eigenfunctions, ortho-normality relation
See also:: Annotations for §1.18(vi), §1.18 and Ch.1

Eigenfunctions corresponding to the continuous spectrum are non- $L^{2}$ functions. Let 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 has no point spectrum, i.e., has no eigenfunctions in $L^{2}\left(X\right)$ , then the spectrum $\boldsymbol{\sigma}$ of 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 , 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 $\int_{0}^{\infty}{\phi_{\lambda}}(x)\overline{\phi_{\lambda^{\prime}}(x)}\,% \mathrm{d}x=\delta\left(\lambda-\lambda^{\prime}\right),$ $\lambda,\lambda^{\prime}\in\boldsymbol{\sigma}_{c}$ ,

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : 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 $\delta\left(x-y\right)=\int_{0}^{\infty}\phi_{\lambda}(x)\overline{\phi_{% \lambda}(y)}\,\mathrm{d}\lambda.$

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of 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 $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$ , 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 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 $\int_{0}^{\infty}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\int_{0}^{\infty}{\left|% \widehat{f}(\lambda)\right|}^{2}\,\mathrm{d}\lambda.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $(Tf)(x)=\int_{0}^{\infty}\left(\int_{0}^{\infty}\lambda\phi_{\lambda}(x)% \overline{\phi_{\lambda}(y)}\,\mathrm{d}\lambda\right)f(y)\,\mathrm{d}y,$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of 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 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 $\left\langle F(T)f,f\right\rangle=\int_{0}^{\infty}F(\lambda){\left|\widehat{f% }(\lambda)\right|}^{2}\,\mathrm{d}\lambda.$

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $F(\lambda)=\left(z-\lambda\right)^{-1}$ , $z\in\mathbb{C}\setminus\boldsymbol{\sigma}_{c}$

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

ⓘ

Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral, $\notin$ : not an element of, : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\lim_{\nu\to 0+}\frac{1}{2\pi\mathrm{i}}\left(\left\langle\left(\mu-\mathrm{i}% \nu-T\right)^{-1}f,f\right\rangle-\left\langle\left(\mu+\mathrm{i}\nu-T\right)% ^{-1}f,f\right\rangle\right)$

ⓘ

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value
Permalink:: http://dlmf.nist.gov/1.18.E54
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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

ⓘ

Keywords:: Bessel eigenfunction expansion, Hankel transform
See also:: Annotations for §1.18(vi), §1.18 and Ch.1

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\Re$ : real part
Permalink:: http://dlmf.nist.gov/1.18.E56
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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

1.18.57 $\lim_{R\to\infty}\int_{0}^{\infty}f(y)\left(\int_{0}^{R}\sqrt{xt}J_{\nu}\left(% xt\right)\sqrt{yt}J_{\nu}\left(yt\right)\,\mathrm{d}t\right)\,\mathrm{d}y=% \tfrac{1}{2}\left(f(x+)+f(x-)\right),$

, $\nu>-1$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of 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

$\int_{1}^{\infty}y^{-1}\left|f(y)\right|\,\mathrm{d}y<\infty,$

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

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.18.E58
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

(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, $X=[0,\infty)$

ⓘ

Keywords:: cosine transform, eigenfunction expansion, sine transform
See also:: Annotations for §1.18(vi), §1.18 and Ch.1

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

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line and $\sin\NVar{z}$ : sine function
Source:: Titchmarsh (1962a, pp. 71–72)
Permalink:: http://dlmf.nist.gov/1.18.E59
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

For even in this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for 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

ⓘ

Keywords:: Anderson localization, almost Mathiew equation, band spectra, random potentials, second order differential operators, singular continuous spectra
Referenced by:: §1.18(vi)
Permalink:: http://dlmf.nist.gov/1.18.vii
See also:: Annotations for §1.18 and Ch.1

More generally, continuous spectra may occur in sets of disjoint finite intervals $[\lambda_{a},\lambda_{b}]\in(0,\infty)$ , often called bands, when is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). Should 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

ⓘ

Keywords:: analytic continuation of matrix elements of the resolvent onto higher Riemann sheets, analytic continuation onto higher Riemann sheets, completeness relation, completness relation, dilatation transformations, eigenfunction expansions, eigenfunctions, matrix elements of the resolvent, mixed spectra, ortho-normality relation
Referenced by:: §1.18, §18.39(i), §18.39(ii), §18.39(ii)
Permalink:: http://dlmf.nist.gov/1.18.viii
See also:: Annotations for §1.18 and Ch.1

In general, operators being formally self-adjoint second order differential operators of the form (1.18.28), with 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 $\int_{X}\phi_{\lambda}(x)\overline{\phi_{\lambda^{\prime}}(x)}\,\mathrm{d}x=% \delta\left(\lambda-\lambda^{\prime}\right),$ $\lambda,\lambda^{\prime}\in\boldsymbol{\sigma}_{c}$ ,

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.18.E60
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

1.18.61 $\int_{X}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{m}}(x)}\,\mathrm{d}x=% \delta_{n,m},$ $\lambda_{n},\lambda_{m}\in\boldsymbol{\sigma}_{p}$ ,

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, : nonnegative integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.18.E61
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

1.18.62 $\int_{X}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda}(x)}\,\mathrm{d}x=0,$ $\lambda_{n}\in\boldsymbol{\sigma}_{p}$ , $\lambda\in\boldsymbol{\sigma}_{c}$ ,

ⓘ

Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.18.E62
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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

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

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.18.E63
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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

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

ⓘ

Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, $\cap$ : intersection, $C(\NVar{I})$ or $C(\NVar{a,b})$ : continuous on an interval or and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.18.E64
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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

ⓘ

Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.18(viii)
Permalink:: http://dlmf.nist.gov/1.18.E65
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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

ⓘ

Symbols:: $L^{2}\left(\NVar{X},\NVar{\,\mathrm{d}x}\right)$ : Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\left\langle\NVar{f},\NVar{g}\right\rangle$ : inner product over functions, $\int$ : integral, $\notin$ : not an element of, : variable, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.18(viii), §1.18(viii)
Permalink:: http://dlmf.nist.gov/1.18.E66
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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 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 with a ‘Dip, or Well’ has a partly discrete spectrum

ⓘ

Keywords:: discrete spectra, in one and two dimensions
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

Suppose that is the whole real line in one dimension, and that , 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 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 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 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 , 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

ⓘ

Keywords:: Schrödinger Operators, Schrödinger–Coulomb problem, mixed spectra
See also:: Annotations for §1.18(viii), §1.18 and Ch.1

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 with respect to 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 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 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, , of discrete states depends on the nature of , as well as $\ell$ , and, again, 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

ⓘ

Referenced by:: §1.18(iii), §1.18(iii), §1.18(vi), §1.18, §10.22(v)
Permalink:: http://dlmf.nist.gov/1.18.ix
See also:: Annotations for §1.18 and Ch.1

Self-Adjoint Operators

ⓘ

Keywords:: essentially self-adjoint operator, mathematical background, range and domain, self-adjoint operator
See also:: Annotations for §1.18(ix), §1.18 and Ch.1

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

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

ⓘ

Symbols:: ${\NVar{\mathbf{A}}}^{*}$ : adjoint of matrix, $\in$ : element of, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\sup$ : least upper bound (supremum), $\left\|{\NVar{\mathbf{v}}}\right\|$ : vector norm (²), : variable and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Notes:: See Dunford and Schwartz (1988, §XII.1.4).
Permalink:: http://dlmf.nist.gov/1.18.E68
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

such that

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

ⓘ

Symbols:: ${\NVar{\mathbf{A}}}^{*}$ : adjoint of matrix, $\in$ : element of, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and : variable
Permalink:: http://dlmf.nist.gov/1.18.E69
Encodings:: TeX, pMML, png
See also:: Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

A linear operator 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 : 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 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 is essentially self-adjoint and if $T={T}^{*}$ then is self-adjoint.

Spectrum of an Operator

ⓘ

Keywords:: mathematical background, spectrum of an operator
Notes:: See Dunford and Schwartz (1988, §XII.1.1) for the spectrum of an operator.
See also:: Annotations for §1.18(ix), §1.18 and Ch.1

Let be a linear operator on with dense domain $\mathcal{D}(T)$ and with range $\mathcal{R}(T)=\{Tv\mid v\in\mathcal{D}(T)\}$ . Such an operator is called injective if, for any in its domain, implies that . The resolvent set $\rho(T)$ consists of all $z\in\mathbb{C}$ such that (i) is injective, (ii) $\mathcal{R}(z-T)$ is dense in , (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.

The point spectrum $\boldsymbol{\sigma}_{p}$ . It consists of all $z\in\mathbb{C}$ for which is not injective, or equivalently, for which is an eigenvalue of , i.e., for some $v\in\mathcal{D}(T)\backslash\{0\}$ .
2.

The continuous spectrum $\boldsymbol{\sigma}_{c}$ . It consists of all $z\in\mathbb{C}$ for which is injective and has dense range, but $\left(T-z\right)^{-1}$ is not bounded.
3.

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

If is a bounded operator then its spectrum is a closed bounded subset of $\mathbb{C}$ . If 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

ⓘ

Keywords:: deficiency indices, mathematical background, self-adjoint extensions of a symmetric operator
Notes:: For self-adjoint extensions of a symmetric operator see Dunford and Schwartz (1988, §XII.4.1–19).
See also:: Annotations for §1.18(ix), §1.18 and Ch.1

Let be a symmetric operator on a Hilbert space , so $\mathcal{D}(T)$ is dense in and $T\subset T^{**}\subset{T}^{*}$ . For $z\in\mathbb{C}\backslash\mathbb{R}$ let $N_{z}$ be the -eigenspace of ${T}^{*}$ , i.e., $N_{z}$ is the linear subspace of $\mathcal{D}({T}^{*})$ consisting of all 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 . Then 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 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

ⓘ

Keywords:: boundary conditions, boundary conditions and the Weyl alternative, limit circle, limit point, limit point and limit circle boundary conditions, mathematical background, second order linear differential operator
Notes:: For the Weyl alternative see Dunford and Schwartz (1988, p. 1306).
See also:: Annotations for §1.18(ix), §1.18 and Ch.1

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 is an ordinary solution (i.e., $f\in C^{2}(X)$ ) of which is moreover in $L^{2}\left(X\right)$ . Thus $N_{z}$ has dimension 0, 1 or 2. Also, because 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 , and $n_{2},n_{2}$ the ones for $\mathcal{L}$ restricted to . Then $n_{1}$ and $n_{2}$ are independent of . 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 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) §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 , and where the limit has to exist for all . Then, if the linear form $\mathcal{B}$ is nonzero, the condition $\mathcal{B}(f)=0$ is called a boundary condition at . Boundary values and boundary conditions for the end point are defined in a similar way. If $n_{1}=1$ then there are no nonzero boundary values at ; if $n_{1}=2$ then the above boundary values at form a two-dimensional class. Similarly at . 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 and, if $n_{2}=2$ , $\mathcal{B}_{2}(f)=0$ for a chosen $\mathcal{B}_{2}$ at .

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

Spectral expansions and self-adjoint extensions

ⓘ

See also:: Annotations for §1.18(ix), §1.18 and Ch.1

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

ⓘ

Referenced by:: Erratum (V1.2.0) §1.18, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.18.x
See also:: Annotations for §1.18 and Ch.1

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.

1.17 Integral and Series Representations of the Dirac Delta 2 Asymptotic Approximations

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§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

Contents

§1.18(i) Hilbert spaces

§1.18(ii) spaces on intervals in

§1.18(iii) Linear Operators on a Hilbert Space

Bounded and Unbounded Linear Operators

Self-Adjoint and Symmetric Operators

Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions

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

§1.18(v) Point Spectra and Eigenfunction Expansions

General Results

Example 1: Three Simple Cases where ,

Hermite’s Differential Equation,

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

General Results

Example 1: Bessel–Hankel Transform,

Example 2: Sine and Cosine Transforms,

§1.18(vii) Continuous Spectra: More General Cases

§1.18(viii) Mixed Spectra and Eigenfunction Expansions

Example 1: In one and two dimensions any with a ‘Dip, or Well’ has a partly discrete spectrum

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

§1.18(ix) Mathematical Background

Self-Adjoint Operators

Spectrum of an Operator

Self-adjoint extensions of a symmetric Operator

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

Spectral expansions and self-adjoint extensions

§1.18(x) Literature

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

Example 1: Three Simple Cases where , $X=[0,\pi]$

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

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

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