Jump to content

Main menu Navigation ●Main page ●Contents ●Current events ●Random article ●About Wikipedia ●Contact us ●Donate Contribute ●Help ●Learn to edit ●Community portal ●Recent changes ●Upload file

●Create account ●Log in ●Create account ● Log in Pages for logged out editors learn more ●Contributions ●Talk

(Top) 1 Introduction 2 Solutions of ordinary differential equations 2.1 Reduction to standard form 2.2 Existence theorem 2.3 Fundamental eigenfunctions 2.4 Green's formula 3 Classical Sturm–Liouville theory 3.1 Green's function (regular case) 3.2 Spectral theorem 3.3 Wronskian as a Fredholm determinant 4 Tools from abstract spectral theory 4.1 Functions of bounded variation 4.2 Spectral measure 5 Weyl–Titchmarsh–Kodaira theory 5.1 Limit circle and limit point for singular equations 5.2 Green's function (singular case) 5.3 Spectral theorem and Titchmarsh–Kodaira formula 6 Application to the hypergeometric equation 7 Application to the hydrogen atom 8 Generalisations and alternative approaches 9 Gelfand–Levitan theory 10 Notes 11 References 11.1 Citations 11.2 Bibliography

Spectral theory of ordinary differential equations

Add links ●Article ●Talk ●Read ●Edit ●View history Tools Actions ●Read ●Edit ●View history General ●What links here ●Related changes ●Upload file ●Special pages ●Permanent link ●Page information ●Cite this page ●Get shortened URL ●Download QR code ●Wikidata item Print/export ●Download as PDF ●Printable version Appearance From Wikipedia, the free encyclopedia

Inmathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysisonsemisimple Lie groups.

Introduction[edit]

Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language, it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles.

In the 1920s, John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the spectral measure. The same formula had also been obtained independently by E. C. Titchmarsh in 1946 (scientific communication between Japan and the United Kingdom had been interrupted by World War II). Titchmarsh had followed the method of the German mathematician Emil Hilb, who derived the eigenfunction expansions using complex function theory instead of operator theory. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the resolvent of the singular differential operator could be approximated by compact resolvents corresponding to Sturm–Liouville problems for proper subintervals. Another method was found by Mark Grigoryevich Krein; his use of direction functionals was subsequently generalised by Izrail Glazman to arbitrary ordinary differential equations of even order.

Weyl applied his theory to Carl Friedrich Gauss's hypergeometric differential equation, thus obtaining a far-reaching generalisation of the transform formula of Gustav Ferdinand Mehler (1881) for the Legendre differential equation, rediscovered by the Russian physicist Vladimir Fock in 1943, and usually called the Mehler–Fock transform. The corresponding ordinary differential operator is the radial part of the Laplacian operator on 2-dimensional hyperbolic space. More generally, the Plancherel theorem for SL(2,R)ofHarish Chandra and Gelfand–Naimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of spherical functions for the isometry groups of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real semisimple Lie groups was strongly influenced by the methods Weyl developed for eigenfunction expansions associated with singular ordinary differential equations. Equally importantly the theory also laid the mathematical foundations for the analysis of the Schrödinger equation and scattering matrixinquantum mechanics.

Solutions of ordinary differential equations[edit]

Reduction to standard form[edit]

Let $D$ be the second order differential operator on $(a, b)$ given by

{\displaystyle Df(

Df(x)=-p(x)f''(x)+r(x)f'(x)+q(x)f(x),

where

p

is a strictly positive continuously differentiable function and

q

and

r

are continuous real-valued functions.

For $x 0$ in $(a, b)$ , define the Liouville transformation $ψ$ by

{\displaystyle \psi (

\psi (x)=\int _{x_{0}}^{x}p(t)^{-1/2}\,dt

{\displaystyle U:L^{2}(a,b)\mapsto L^{2}(\psi (

U:L^{2}(a,b)\mapsto L^{2}(\psi (a),\psi (b))

is the unitary operator defined by

{\displaystyle (

(Uf)(\psi (x))=f(x)\times \left(\psi '(x)\right)^{-1/2},\ \ \forall x\in (a,b)

then

U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}g=g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}

U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}g=g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}

and

{\begin{aligned}U{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}U^{-1}g&=\left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)\times \left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)g\\[1ex]&={\frac {\mathrm {d} }{\mathrm {d} \psi }}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot \psi '+{\frac {1}{2}}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot {\frac {\psi ''}{\psi '}}\\[1ex]&=g''\psi '^{2}+2g'\psi ''+{\frac {1}{2}}g\cdot \left[{\frac {\psi '''}{\psi '}}-{\frac {1}{2}}{\frac {\psi ''^{2}}{\psi '^{2}}}\right]\end{aligned}}

{\begin{aligned}U{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}U^{-1}g&=\left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)\times \left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)g\\[1ex]&={\frac {\mathrm {d} }{\mathrm {d} \psi }}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot \psi '+{\frac {1}{2}}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot {\frac {\psi ''}{\psi '}}\\[1ex]&=g''\psi '^{2}+2g'\psi ''+{\frac {1}{2}}g\cdot \left[{\frac {\psi '''}{\psi '}}-{\frac {1}{2}}{\frac {\psi ''^{2}}{\psi '^{2}}}\right]\end{aligned}}

Hence,

UDU^{-1}g=-g''+Rg'+Qg,

UDU^{-1}g=-g''+Rg'+Qg,

where

R={\frac {p'+r}{p^{1/2}}}

R={\frac {p'+r}{p^{1/2}}}

and

Q=q-{\frac {rp'}{4p}}+{\frac {p''}{4}}-{\frac {5p'^{2}}{16p}}

Q=q-{\frac {rp'}{4p}}+{\frac {p''}{4}}-{\frac {5p'^{2}}{16p}}

The term in $g'$ can be removed using an Euler integrating factor. If $S' / S = - R /2$ , then $h = Sg$ satisfies

(SUDU^{-1}S^{-1})h=-h''+Vh,

(SUDU^{-1}S^{-1})h=-h''+Vh,

where the potential V is given by

V=Q+{\frac {S''}{S}}

V=Q+{\frac {S''}{S}}

The differential operator can thus always be reduced to one of the form ^[1]

Df=-f''+qf.

Df=-f''+qf.

Existence theorem[edit]

The following is a version of the classical Picard existence theorem for second order differential equations with values in a Banach space $E$ .^[2]

Let $α$ , $β$ be arbitrary elements of $E$ , $A$ abounded operatoron $E$ and $q$ a continuous function on $[a, b]$ .

Then, for $c = a$ or $c = b$ , the differential equation

Df=Af

Df=Af

has a unique solution

f

C 2 ([a, b], E)

satisfying the initial conditions

{\displaystyle f(

f(c)=\beta \,,\;f'(c)=\alpha .

In fact a solution of the differential equation with these initial conditions is equivalent to a solution of the integral equation

f=h+Tf

f=h+Tf

with

T

the bounded linear map on

C ([a, b], E)

defined by

{\displaystyle Tf(

Tf(x)=\int _{c}^{x}K(x,y)f(y)\,dy,

where

K

is the Volterra kernel

{\displaystyle K(x,t)=(x-t)(q(

K(x,t)=(x-t)(q(t)-A)

and

{\displaystyle h(

h(x)=\alpha (x-c)+\beta .

Since $‖ T k ‖$ tends to 0, this integral equation has a unique solution given by the Neumann series

f=(I-T)^{-1}h=h+Th+T^{2}h+T^{3}h+\cdots

f=(I-T)^{-1}h=h+Th+T^{2}h+T^{3}h+\cdots

This iterative scheme is often called Picard iteration after the French mathematician Charles Émile Picard.

Fundamental eigenfunctions[edit]

If $f$ is twice continuously differentiable (i.e. $C 2$ ) on $(a, b)$ satisfying $Df = λf$ , then $f$ is called an eigenfunctionof $D$ with eigenvalue $λ$ .

In the case of a compact interval $[a, b]$ and $q$ continuous on $[a, b]$ , the existence theorem implies that for $c = a$ or $c = b$ and every complex number $λ$ there a unique $C 2$ eigenfunction $f λ$ on $[a, b]$ with $f λ (c)$ and $f' λ (c)$ prescribed. Moreover, for each $x$ in $[a, b]$ , $f λ (x)$ and $f' λ (x)$ are holomorphic functionsof $λ$ .
For an arbitrary interval $(a, b)$ and $q$ continuous on $(a, b)$ , the existence theorem implies that for $c$ in $(a, b)$ and every complex number $λ$ there a unique $C 2$ eigenfunction $f λ$ on $(a, b)$ with $f λ (c)$ and $f' λ (c)$ prescribed. Moreover, for each $x$ in $(a, b)$ , $f λ (x)$ and $f' λ (x)$ are holomorphic functionsof $λ$ .

Green's formula[edit]

If $f$ and $g$ are $C 2$ functions on $(a, b)$ , the Wronskian $W (f, g)$ is defined by

{\displaystyle W(f,g)(

W(f,g)(x)=f(x)g'(x)-f'(x)g(x).

Green's formula - which in this one-dimensional case is a simple integration by parts - states that for $x$ , $y$ in $(a, b)$

{\displaystyle \int _{x}^{y}(

\int _{x}^{y}(Df)g-f(Dg)\,dt=W(f,g)(y)-W(f,g)(x).

When $q$ is continuous and $f$ , $g$ are $C 2$ on the compact interval $[a, b]$ , this formula also holds for $x = a$ or $y = b$ .

When $f$ and $g$ are eigenfunctions for the same eigenvalue, then

{\frac {d}{dx}}W(f,g)=0,

{\frac {d}{dx}}W(f,g)=0,

so that

W (f, g)

is independent of

x

Classical Sturm–Liouville theory[edit]

Let $[a, b]$ be a finite closed interval, $q$ a real-valued continuous function on $[a, b]$ and let $H 0$ be the space of $C 2$ functions $f$ on $[a, b]$ satisfying the Robin boundary conditions

{\displaystyle {\begin{cases}\cos \alpha \,f(

{\begin{cases}\cos \alpha \,f(a)-\sin \alpha \,f'(a)=0,\\[0.5ex]\cos \beta \,f(b)-\sin \beta \,f'(b)=0,\end{cases}}

with inner product

{\displaystyle (f,g)=\int _{a}^{b}f(

(f,g)=\int _{a}^{b}f(x){\overline {g(x)}}\,dx.

In practice usually one of the two standard boundary conditions:

Dirichlet boundary condition $f (c) = 0$
Neumann boundary condition $f'(c) = 0$

is imposed at each endpoint $c = a, b$ .

The differential operator $D$ given by

Df=-f''+qf

Df=-f''+qf

acts on

H 0

. A function

f

H 0

is called an eigenfunctionof

D

(for the above choice of boundary values) if

Df = λ f

for some complex number

λ

, the corresponding eigenvalue. By Green's formula,

D

is formally self-adjointon

H 0

, since the Wronskian

W (f, g)

vanishes if both

f

g

satisfy the boundary conditions:

(Df,g)=(f,Dg),\quad {\text{ for }}f,g\in H_{0}.

(Df,g)=(f,Dg),\quad {\text{ for }}f,g\in H_{0}.

As a consequence, exactly as for a self-adjoint matrix in finite dimensions,

the eigenvalues of $D$ are real;
the eigenspaces for distinct eigenvalues are orthogonal.

It turns out that the eigenvalues can be described by the maximum-minimum principle of Rayleigh–Ritz^[3] (see below). In fact it is easy to see a priori that the eigenvalues are bounded below because the operator $D$ is itself bounded belowon $H 0$ :

(Df,f)\geq M(f,f)

for some finite (possibly negative) constant

M

In fact, integrating by parts,

(Df,f)=\left[-f'{\overline {f}}\right]_{a}^{b}+\int |f'|^{2}+\int q|f|^{2}.

(Df,f)=\left[-f'{\overline {f}}\right]_{a}^{b}+\int |f'|^{2}+\int q|f|^{2}.

For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with $M = inf q$ .

For general Robin boundary conditions the first term can be estimated using an elementary Peter-Paul version of Sobolev's inequality:

"Given $ε > 0$ , there is constant $R > 0$ such that $| f (x) | 2 \leq ε (f', f') + R (f, f)$ for all $f$ in $C 1 [a, b]$ ."

In fact, since

{\displaystyle |f(

|f(b)-f(x)|\leq (b-a)^{1/2}\cdot \|f'\|_{2},

only an estimate for

f (b)

is needed and this follows by replacing

f (x)

in the above inequality by

(x - a) n \cdot(b - a) - n \cdot f (x)

for

n

sufficiently large.

Green's function (regular case)[edit]

From the theory of ordinary differential equations, there are unique fundamental eigenfunctions $φ λ (x)$ , $χ λ (x)$ such that

$D φ λ = λ φ λ$ , $φ λ (a) = sin α$ , $φ λ'(a) = cos α$
$D χ λ = λ χ λ$ , $χ λ (b) = sin β$ , $χ λ'(b) = cos β$

which at each point, together with their first derivatives, depend holomorphically on $λ$ . Let

\omega (\lambda )=W(\phi _{\lambda },\chi _{\lambda }),

\omega (\lambda )=W(\phi _{\lambda },\chi _{\lambda }),

be an entire holomorphic function.

This function $ω (λ)$ plays the role of the characteristic polynomialof $D$ . Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of $D$ and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of $D$ and, if there are infinitely many, they must tend to infinity. It turns out that the zeros of $ω (λ)$ also have mutilplicity one (see below).

If $λ$ is not an eigenvalue of $D$ on $H 0$ , define the Green's functionby

{\displaystyle G_{\lambda }(x,y)={\begin{cases}\phi _{\lambda }(

G_{\lambda }(x,y)={\begin{cases}\phi _{\lambda }(x)\chi _{\lambda }(y)/\omega (\lambda )&{\text{ for }}x\geq y\\[1ex]\chi _{\lambda }(x)\phi _{\lambda }(y)/\omega (\lambda )&{\text{ for }}y\geq x.\end{cases}}

This kernel defines an operator on the inner product space $C [a, b]$ via

{\displaystyle (G_{\lambda }f)(

(G_{\lambda }f)(x)=\int _{a}^{b}G_{\lambda }(x,y)f(y)\,dy.

Since $G λ (x, y)$ is continuous on $[a, b] \times [a, b]$ , it defines a Hilbert–Schmidt operator on the Hilbert space completion $H$ of $C [a, b] = H 1$ (or equivalently of the dense subspace $H 0$ ), taking values in $H 1$ . This operator carries $H 1$ into $H 0$ . When $λ$ is real, $G λ (x, y) = G λ (y, x)$ is also real, so defines a self-adjoint operator on $H$ . Moreover,

$G λ (D - λ) = I$ on $H 0$
$G λ$ carries $H 1$ into $H 0$ , and $(D - λ) G λ = I$ on $H 1$ .

Thus the operator $G λ$ can be identified with the resolvent $(D - λ) -1$ .

Spectral theorem[edit]

Theorem — The eigenvalues of D are real of multiplicity one and form an increasing sequence $λ 1 < λ 2 < \dots$ tending to infinity.

The corresponding normalised eigenfunctions form an orthonormal basis of $H 0$ .

The $k$ -th eigenvalue of $D$ is given by the minimax principle

\lambda _{k}=\max _{\dim G=k-1}\,\min _{f\perp G}{(Df,f) \over (f,f)}.

\lambda _{k}=\max _{\dim G=k-1}\,\min _{f\perp G}{(Df,f) \over (f,f)}.

In particular if $q 1 \leq q 2$ , then

\lambda _{k}(D_{1})\leq \lambda _{k}(D_{2}).

\lambda _{k}(D_{1})\leq \lambda _{k}(D_{2}).

In fact let $T = G λ$ for $λ$ large and negative. Then $T$ defines a compact self-adjoint operator on the Hilbert space $H$ . By the spectral theorem for compact self-adjoint operators, $H$ has an orthonormal basis consisting of eigenvectors $ψ n$ of $T$ with $Tψ n = μ n ψ n$ , where $μ n$ tends to zero. The range of $T$ contains $H 0$ so is dense. Hence 0 is not an eigenvalue of $T$ . The resolvent properties of $T$ imply that $ψ n$ lies in $H 0$ and that

D\psi _{n}=\left(\lambda +{\frac {1}{\mu _{n}}}\right)\psi _{n}

D\psi _{n}=\left(\lambda +{\frac {1}{\mu _{n}}}\right)\psi _{n}

The minimax principle follows because if

{\displaystyle \lambda (

\lambda (G)=\min _{f\perp G}{\frac {(Df,f)}{(f,f)}},

then

λ (G) = λ k

for the linear span of the first

k - 1

eigenfunctions. For any other

(k - 1)

-dimensional subspace

G

, some

f

in the linear span of the first

k

eigenvectors must be orthogonal to

G

. Hence

λ (G) \leq (Df, f)/(f, f) \leq λ k

Wronskian as a Fredholm determinant[edit]

For simplicity, suppose that $m \leq q (x) \leq M$ on $[0, π]$ with Dirichlet boundary conditions. The minimax principle shows that

{\displaystyle n^{2}+m\leq \lambda _{n}(

n^{2}+m\leq \lambda _{n}(D)\leq n^{2}+M.

It follows that the resolvent $(D - λ) -1$ is a trace-class operator whenever $λ$ is not an eigenvalue of $D$ and hence that the Fredholm determinant $det I - μ (D - λ) -1$ is defined.

The Dirichlet boundary conditions imply that

{\displaystyle \omega (\lambda )=\phi _{\lambda }(

\omega (\lambda )=\phi _{\lambda }(b).

Using Picard iteration, Titchmarsh showed that $φ λ (b)$ , and hence $ω (λ)$ , is an entire function of finite order $1/2$ :

\omega (\lambda )={\mathcal {O}}\left(e^{\sqrt {|\lambda |}}\right)

\omega (\lambda )={\mathcal {O}}\left(e^{\sqrt {|\lambda |}}\right)

At a zero $μ$ of $ω (λ)$ , $φ μ (b) = 0$ . Moreover,

{\displaystyle \psi (

\psi (x)=\partial _{\lambda }\varphi _{\lambda }(x)|_{\lambda =\mu }

satisfies

(D - μ) ψ = φ μ

. Thus

{\displaystyle \omega (\lambda )=(\lambda -\mu )\psi (

\omega (\lambda )=(\lambda -\mu )\psi (b)+{\mathcal {O}}((\lambda -\mu )^{2})

This implies that^[4]

μ

is a simple zero of

ω (λ)

For otherwise $ψ (b) = 0$ , so that $ψ$ would have to lie in $H 0$ . But then

(\phi _{\mu },\phi _{\mu })=((D-\mu )\psi ,\phi _{\mu })=(\psi ,(D-\mu )\phi _{\mu })=0,

(\phi _{\mu },\phi _{\mu })=((D-\mu )\psi ,\phi _{\mu })=(\psi ,(D-\mu )\phi _{\mu })=0,

a contradiction.

On the other hand, the distribution of the zeros of the entire function ω(λ) is already known from the minimax principle.

By the Hadamard factorization theorem, it follows that^[5]

\omega (\lambda )=C\prod (1-\lambda /\lambda _{n}),

\omega (\lambda )=C\prod (1-\lambda /\lambda _{n}),

for some non-zero constant

C

Hence

\det(I-\mu (D-\lambda )^{-1})=\prod \left(1-{\mu  \over \lambda _{n}-\lambda }\right)=\prod {1-(\lambda +\mu )/\lambda _{n} \over 1-\lambda /\lambda _{n}}={\omega (\lambda +\mu ) \over \omega (\lambda )}.

\det(I-\mu (D-\lambda )^{-1})=\prod \left(1-{\mu  \over \lambda _{n}-\lambda }\right)=\prod {1-(\lambda +\mu )/\lambda _{n} \over 1-\lambda /\lambda _{n}}={\omega (\lambda +\mu ) \over \omega (\lambda )}.

In particular if 0 is not an eigenvalue of $D$

\omega (\mu )=\omega (0)\cdot \det(I-\mu D^{-1}).

\omega (\mu )=\omega (0)\cdot \det(I-\mu D^{-1}).

Tools from abstract spectral theory[edit]

Functions of bounded variation[edit]

A function $ρ (x)$ ofbounded variation^[6] on a closed interval $[a, b]$ is a complex-valued function such that its total variation $V (ρ)$ , the supremum of the variations

\sum _{r=0}^{k-1}|\rho (x_{r+1})-\rho (x_{r})|

\sum _{r=0}^{k-1}|\rho (x_{r+1})-\rho (x_{r})|

over all dissections

a=x_{0}<x_{1}<\dots <x_{k}=b

a=x_{0}<x_{1}<\dots <x_{k}=b

is finite. The real and imaginary parts of

ρ

are real-valued functions of bounded variation. If

ρ

is real-valued and normalised so that

ρ (a) = 0

, it has a canonical decomposition as the difference of two bounded non-decreasing functions:

{\displaystyle \rho (

\rho (x)=\rho _{+}(x)-\rho _{-}(x),

where

ρ + (x)

and

ρ - (x)

are the total positive and negative variation of

ρ

over

[a, x]

If $f$ is a continuous function on $[a, b]$ its Riemann–Stieltjes integral with respect to $ρ$

{\displaystyle \int _{a}^{b}f(

\int _{a}^{b}f(x)\,d\rho (x)

is defined to be the limit of approximating sums

\sum _{r=0}^{k-1}f(x_{r})(\rho (x_{r+1})-\rho (x_{r}))

\sum _{r=0}^{k-1}f(x_{r})(\rho (x_{r+1})-\rho (x_{r}))

as the mesh of the dissection, given by

sup | x r +1 - x r |

, tends to zero.

This integral satisfies

{\displaystyle \left|\int _{a}^{b}f(

\left|\int _{a}^{b}f(x)\,d\rho (x)\right|\leq V(\rho )\cdot \|f\|_{\infty }

and thus defines a bounded linear functional $dρ$ on $C [a, b]$ with norm $‖ dρ ‖ = V (ρ)$ .

Every bounded linear functional $μ$ on $C [a, b]$ has an absolute value |μ| defined for non-negative $f$ by^[7]

{\displaystyle |\mu |(

|\mu |(f)=\sup _{0\leq |g|\leq f}|\mu (g)|.

The form $| μ |$ extends linearly to a bounded linear form on $C [a, b]$ with norm $‖ μ ‖$ and satisfies the characterizing inequality

{\displaystyle |\mu (

|\mu (f)|\leq |\mu |(|f|)

for

f

C [a, b]

. If

μ

isreal, i.e. is real-valued on real-valued functions, then

\mu =|\mu |-(|\mu |-\mu )\equiv \mu _{+}-\mu _{-}

\mu =|\mu |-(|\mu |-\mu )\equiv \mu _{+}-\mu _{-}

gives a canonical decomposition as a difference of positive forms, i.e. forms that are non-negative on non-negative functions.

Every positive form $μ$ extends uniquely to the linear span of non-negative bounded lower semicontinuous functions $g$ by the formula^[8]

{\displaystyle \mu (

\mu (g)=\lim \mu (f_{n}),

where the non-negative continuous functions

f n

increase pointwise to

g

The same therefore applies to an arbitrary bounded linear form $μ$ , so that a function $ρ$ of bounded variation may be defined by^[9]

{\displaystyle \rho (

\rho (x)=\mu (\chi _{[a,x]}),

where

χ A

denotes the characteristic function of a subset

A

[a, b]

. Thus

μ = dρ

and

‖ μ ‖ = ‖ dρ ‖

. Moreover

μ + = dρ +

and

μ - = dρ -

This correspondence between functions of bounded variation and bounded linear forms is a special case of the Riesz representation theorem.

The supportof $μ = dρ$ is the complement of all points $x$ in $[a, b]$ where $ρ$ is constant on some neighborhood of $x$ ; by definition it is a closed subset $A$ of $[a, b]$ . Moreover, $μ ((1 - χ A) f) = 0$ , so that $μ (f) = 0$ if $f$ vanishes on $A$ .

Spectral measure[edit]

Let $H$ be a Hilbert space and $T$ a self-adjoint bounded operatoron $H$ with $0\leq T\leq I$ , so that the spectrum ${\displaystyle \sigma ($ of $T$ is contained in $[0,1]$ . If ${\displaystyle p($ is a complex polynomial, then by the spectral mapping theorem

{\displaystyle \sigma (p(

\sigma (p(T))=p(\sigma (T))

and hence

{\displaystyle \|p(

\|p(T)\|\leq \|p\|_{\infty }

where

\|\cdot \|_{\infty }

denotes the uniform normon

C [0, 1]

. By the Weierstrass approximation theorem, polynomials are uniformly dense in

C [0, 1]

. It follows that

{\displaystyle f(

can be defined

\forall f\in C[0,1]

, with

{\displaystyle \sigma (f(

\sigma (f(T))=f(\sigma (T))

and

{\displaystyle \|f(

\|f(T)\|\leq \|f\|_{\infty }.

If $0\leq g\leq 1$ is a lower semicontinuous function on $[0, 1]$ , for example the characteristic function $\chi _{[0,\alpha ]}$ of a subinterval of $[0, 1]$ , then $g$ is a pointwise increasing limit of non-negative $f_{n}\in C[0,1]$ .

If $\xi$ is a vector in $H$ , then the vectors

{\displaystyle \eta _{n}=f_{n}(

\eta _{n}=f_{n}(T)\xi

form a Cauchy sequencein

H

, since, for

n\geq m

\|\eta _{n}-\eta _{m}\|^{2}\leq (\eta _{n},\xi )-(\eta _{m},\xi ),

\|\eta _{n}-\eta _{m}\|^{2}\leq (\eta _{n},\xi )-(\eta _{m},\xi ),

and

{\displaystyle (\eta _{n},\xi )=(f_{n}(

is bounded and increasing, so has a limit.

It follows that ${\displaystyle g($ can be defined by^[a]

{\displaystyle g(

g(T)\xi =\lim f_{n}(T)\xi .

If $\xi$ and $η$ are vectors in $H$ , then

{\displaystyle \mu _{\xi ,\eta }(

\mu _{\xi ,\eta }(f)=(f(T)\xi ,\eta )

defines a bounded linear form

\mu _{\xi ,\eta }

H

. By the Riesz representation theorem

\mu _{\xi ,\eta }=d\rho _{\xi ,\eta }

\mu _{\xi ,\eta }=d\rho _{\xi ,\eta }

for a unique normalised function

\rho _{\xi ,\eta }

of bounded variation on

[0, 1]

$d\rho _{\xi ,\eta }$ (or sometimes slightly incorrectly $\rho _{\xi ,\eta }$ itself) is called the spectral measure determined by $\xi$ and $η$ .

The operator ${\displaystyle g($ is accordingly uniquely characterised by the equation

{\displaystyle (g(

(g(T)\xi ,\eta )=\mu _{\xi ,\eta }(g)=\int _{0}^{1}g(\lambda )\,d\rho _{\xi ,\eta }(\lambda ).

The spectral projection $E(\lambda )$ is defined by

{\displaystyle E(\lambda )=\chi _{[0,\lambda ]}(

E(\lambda )=\chi _{[0,\lambda ]}(T),

so that

\rho _{\xi ,\eta }(\lambda )=(E(\lambda )\xi ,\eta ).

\rho _{\xi ,\eta }(\lambda )=(E(\lambda )\xi ,\eta ).

It follows that

{\displaystyle g(

g(T)=\int _{0}^{1}g(\lambda )\,dE(\lambda ),

which is understood in the sense that for any vectors

\xi

and

\eta

{\displaystyle (g(

(g(T)\xi ,\eta )=\int _{0}^{1}g(\lambda )\,d(E(\lambda )\xi ,\eta )=\int _{0}^{1}g(\lambda )\,d\rho _{\xi ,\eta }(\lambda ).

For a single vector $\xi ,\,\mu _{\xi }=\mu _{\xi ,\xi }$ is a positive form on $[0, 1]$ (in other words proportional to a probability measureon $[0, 1]$ ) and $\rho _{\xi }=\rho _{\xi ,\xi }$ is non-negative and non-decreasing. Polarisation shows that all the forms $\mu _{\xi ,\eta }$ can naturally be expressed in terms of such positive forms, since

\mu _{\xi ,\eta }={\frac {1}{4}}\left(\mu _{\xi +\eta }+i\mu _{\xi +i\eta }-\mu _{\xi -\eta }-i\mu _{\xi -i\eta }\right)

\mu _{\xi ,\eta }={\frac {1}{4}}\left(\mu _{\xi +\eta }+i\mu _{\xi +i\eta }-\mu _{\xi -\eta }-i\mu _{\xi -i\eta }\right)

If the vector $\xi$ is such that the linear span of the vectors $(T^{n}\xi )$ is dense in $H$ , i.e. $\xi$ is a cyclic vector for $T$ , then the map $U$ defined by

{\displaystyle U(

U(f)=f(T)\xi ,\,C[0,1]\rightarrow H

satisfies

(Uf_{1},Uf_{2})=\int _{0}^{1}f_{1}(\lambda ){\overline {f_{2}(\lambda )}}\,d\rho _{\xi }(\lambda ).

(Uf_{1},Uf_{2})=\int _{0}^{1}f_{1}(\lambda ){\overline {f_{2}(\lambda )}}\,d\rho _{\xi }(\lambda ).

Let $L_{2}([0,1],d\rho _{\xi })$ denote the Hilbert space completion of $C[0,1]$ associated with the possibly degenerate inner product on the right hand side.^[b] Thus $U$ extends to a unitary transformationof $L_{2}([0,1],\rho _{\xi })$ onto $H$ . $UTU^{\ast }$ is then just multiplication by $\lambda$ on $L_{2}([0,1],d\rho _{\xi })$ ; and more generally ${\displaystyle Uf($ is multiplication by $f(\lambda )$ . In this case, the support of $d\rho _{\xi }$ is exactly ${\displaystyle \sigma ($ , so that

the self-adjoint operator becomes a multiplication operator on the space of functions on its spectrum with inner product given by the spectral measure.

Weyl–Titchmarsh–Kodaira theory[edit]

The eigenfunction expansion associated with singular differential operators of the form

Df=-(pf')'+qf

Df=-(pf')'+qf

on an open interval

(a, b)

requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints

a

and

b

to determine possible boundary conditions there. Unlike the regular Sturm–Liouville case, in some circumstances spectral valuesof

D

can have multiplicity 2. In the development outlined below standard assumptions will be imposed on

p

and

q

that guarantee that the spectrum of

D

has multiplicity one everywhere and is bounded below. This includes almost all important applications; modifications required for the more general case will be discussed later.

Having chosen the boundary conditions, as in the classical theory the resolvent of $D$ , $(D + R) -1$ for $R$ large and positive, is given by an operator $T$ corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case $T$ was a compact self-adjoint operator; in this case $T$ is just a self-adjoint bounded operator with $0 \leq T \leq I$ . The abstract theory of spectral measure can therefore be applied to $T$ to give the eigenfunction expansion for $D$ .

The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of $D$ lies in $[1, \infty)$ and that $T = D -1$ and let

{\displaystyle E(\lambda )=\chi _{[\lambda ^{-1},1]}(

E(\lambda )=\chi _{[\lambda ^{-1},1]}(T)

be the spectral projection of

D

corresponding to the interval

[1, λ]

. For an arbitrary function

f

define

{\displaystyle f(x,\lambda )=(E(\lambda )f)(

f(x,\lambda )=(E(\lambda )f)(x).

f (x, λ)

may be regarded as a differentiable map into the space of functions of bounded variation ρ; or equivalently as a differentiable map

{\displaystyle x\mapsto (d_{\lambda }f)(

x\mapsto (d_{\lambda }f)(x)

into the Banach space

E

of bounded linear functionals

dρ

C [α, β]

whenever

[α, β]

is a compact subinterval of

[1, \infty)

Weyl's fundamental observation was that $d λ f$ satisfies a second order ordinary differential equation taking values in $E$ :

D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f.

D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f.

After imposing initial conditions on the first two derivatives at a fixed point $c$ , this equation can be solved explicitly in terms of the two fundamental eigenfunctions and the "initial value" functionals

{\displaystyle (d_{\lambda }f)(

(d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot ),\quad (d_{\lambda }f)^{\prime }(c)=d_{\lambda }f_{x}(c,\cdot ).

This point of view may now be turned on its head: $f (c, λ)$ and $f x (c, λ)$ may be written as

f(c,\lambda )=(f,\xi _{1}(\lambda )),\quad f_{x}(c,\lambda )=(f,\xi _{2}(\lambda )),

f(c,\lambda )=(f,\xi _{1}(\lambda )),\quad f_{x}(c,\lambda )=(f,\xi _{2}(\lambda )),

where

ξ 1 (λ)

and

ξ 2 (λ)

are given purely in terms of the fundamental eigenfunctions. The functions of bounded variation

\sigma _{ij}(\lambda )=(\xi _{i}(\lambda ),\xi _{j}(\lambda ))

\sigma _{ij}(\lambda )=(\xi _{i}(\lambda ),\xi _{j}(\lambda ))

determine a spectral measure on the spectrum of

D

and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the Titchmarsh–Kodaira formula).

Limit circle and limit point for singular equations[edit]

Let $q (x)$ be a continuous real-valued function on $(0, \infty)$ and let $D$ be the second order differential operator

Df=-f''+qf

Df=-f''+qf

(0, \infty)

. Fix a point

c

(0, \infty)

and, for complex

λ

, let

\varphi _{\lambda },\theta _{\lambda }

be the unique fundamental eigenfunctionsof

D

(0, \infty)

satisfying

(D-\lambda )\varphi _{\lambda }=0,\quad (D-\lambda )\theta _{\lambda }=0

(D-\lambda )\varphi _{\lambda }=0,\quad (D-\lambda )\theta _{\lambda }=0

together with the initial conditions at

c

{\displaystyle \varphi _{\lambda }(

\varphi _{\lambda }(c)=1,\,\varphi _{\lambda }'(c)=0,\,\theta _{\lambda }(c)=0,\,\theta _{\lambda }'(c)=1.

Then their Wronskian satisfies

W(\varphi _{\lambda },\theta _{\lambda })=\varphi _{\lambda }\theta _{\lambda }'-\theta _{\lambda }\varphi _{\lambda }'\equiv 1,

W(\varphi _{\lambda },\theta _{\lambda })=\varphi _{\lambda }\theta _{\lambda }'-\theta _{\lambda }\varphi _{\lambda }'\equiv 1,

since it is constant and equal to 1 at $c$ .

Let $λ$ be non-real and $0 < x < \infty$ . If the complex number $\mu$ is such that $f=\varphi +\mu \theta$ satisfies the boundary condition ${\displaystyle \cos \beta \,f($ for some $\beta$ (or, equivalently, ${\displaystyle f'($ is real) then, using integration by parts, one obtains

\operatorname {Im} (\lambda )\int _{c}^{x}|\varphi +\mu \theta |^{2}=\operatorname {Im} (\mu ).

\operatorname {Im} (\lambda )\int _{c}^{x}|\varphi +\mu \theta |^{2}=\operatorname {Im} (\mu ).

Therefore, the set of $μ$ satisfying this equation is not empty. This set is a circle in the complex $μ$ -plane. Points $μ$ in its interior are characterized by

\int _{c}^{x}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}

\int _{c}^{x}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}

x > c

and by

\int _{x}^{c}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}

\int _{x}^{c}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}

x < c

Let $D x$ be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as $x$ approaches $0$ or $\infty$ . So in the limit, the circles tend either to a limit circle or a limit point at each end. If $\mu$ is a limit point or a point on the limit circle at $0$ or $\infty$ , then $f=\varphi +\mu \theta$ issquare integrable ( $L 2$ ) near $0$ or $\infty$ , since $\mu$ lies in $D x$ for all $x > c$ (in the ∞ case) and so $\int _{c}^{x}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}$ is bounded independent of $x$ . In particular:^[10]

there are always non-zero solutions of $Df = λf$ which are square integrable near $0$ resp. $\infty$ ;
in the limit circle case all solutions of $Df = λf$ are square integrable near $0$ resp. $\infty$ .

The radius of the disc $D x$ can be calculated to be

\left|{1 \over {2\operatorname {Im} (\lambda )\int _{c}^{x}|\theta |^{2}}}\right|

\left|{1 \over {2\operatorname {Im} (\lambda )\int _{c}^{x}|\theta |^{2}}}\right|

and this implies that in the limit point case

\theta

cannot be square integrable near

0

resp.

\infty

. Therefore, we have a converse to the second statement above:

in the limit point case there is exactly one non-zero solution (up to scalar multiples) of Df = λf which is square integrable near $0$ resp. $\infty$ .

On the other hand, if $Dg = λ' g$ for another value $λ'$ , then

{\displaystyle h(

h(x)=g(x)-(\lambda ^{\prime }-\lambda )\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))g(y)\,dy

satisfies

Dh = λh

, so that

{\displaystyle g(

g(x)=c_{1}\varphi _{\lambda }+c_{2}\theta _{\lambda }+(\lambda ^{\prime }-\lambda )\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))g(y)\,dy.

This formula may also be obtained directly by the variation of constant method from $(D - λ) g = (λ' - λ) g$ . Using this to estimate $g$ , it follows that^[10]

the limit point/limit circle behaviour at $0$ or $\infty$ is independent of the choice of $λ$ .

More generally if $Dg = (λ - r) g$ for some function $r (x)$ , then^[11]

{\displaystyle g(

g(x)=c_{1}\varphi _{\lambda }+c_{2}\theta _{\lambda }-\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))r(y)g(y)\,dy.

From this it follows that^[11]

if $r$ is continuous at $0$ , then $D + r$ is limit point or limit circle at $0$ precisely when $D$ is,

so that in particular^[12]

if $q (x) - a / x 2$ is continuous at $0$ , then $D$ is limit point at $0$ if and only if $a ≥ .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}3/4$ .

Similarly

if $r$ has a finite limit at $\infty$ , then $D + r$ is limit point or limit circle at $\infty$ precisely when $D$ is,

so that in particular^[13]

if $q$ has a finite limit at $\infty$ , then $D$ is limit point at $\infty$ .

Many more elaborate criteria to be limit point or limit circle can be found in the mathematical literature.

Green's function (singular case)[edit]

Consider the differential operator

D_{0}f=-(p_{0}f')'+q_{0}f

D_{0}f=-(p_{0}f')'+q_{0}f

(0, \infty)

with

q 0

positive and continuous on

(0, \infty)

and

p 0

continuously differentiable in

[0, \infty)

, positive in

(0, \infty)

and

p 0 (0) = 0

Moreover, assume that after reduction to standard form $D 0$ becomes the equivalent operator

Df=-f''+qf

Df=-f''+qf

(0, \infty)

where

q

has a finite limit at

\infty

. Thus

$D$ is limit point at $\infty$ .

At 0, $D$ may be either limit circle or limit point. In either case there is an eigenfunction $Φ 0$ with $D Φ 0 = 0$ and $Φ 0$ square integrable near $0$ . In the limit circle case, $Φ 0$ determines a boundary conditionat $0$ :

W(f,\Phi _{0})(0)=0.

W(f,\Phi _{0})(0)=0.

For complex $λ$ , let $Φ λ$ and $Χ λ$ satisfy

$(D - λ)Φ λ = 0$ , $(D - λ)Χ λ = 0$
$Χ λ$ square integrable near infinity
$Φ λ$ square integrable at $0$ if $0$ islimit point
$Φ λ$ satisfies the boundary condition above if $0$ islimit circle.

Let

\omega (\lambda )=W(\Phi _{\lambda },\mathrm {X} _{\lambda }),

\omega (\lambda )=W(\Phi _{\lambda },\mathrm {X} _{\lambda }),

a constant which vanishes precisely when

Φ λ

and

Χ λ

are proportional, i.e.

λ

is an eigenvalueof

D

for these boundary conditions.

On the other hand, this cannot occur if $Im λ \neq 0$ or if $λ$ is negative.^[10]

Indeed, if $D f = λf$ with $q 0 - λ \geq δ > 0$ , then by Green's formula $(Df, f) = (f, Df)$ , since $W (f, f *)$ is constant. So $λ$ must be real. If $f$ is taken to be real-valued in the $D 0$ realization, then for $0 < x < y$

[p_{0}ff']_{x}^{y}=\int _{x}^{y}(q_{0}-\lambda )|f|^{2}+p_{0}(f')^{2}.

[p_{0}ff']_{x}^{y}=\int _{x}^{y}(q_{0}-\lambda )|f|^{2}+p_{0}(f')^{2}.

Since $p 0 (0) = 0$ and $f$ is integrable near $0$ , $p 0 f f'$ must vanish at $0$ . Setting $x = 0$ , it follows that $f (y) f'(y) > 0$ , so that $f 2$ is increasing, contradicting the square integrability of $f$ near $\infty$ .

Thus, adding a positive scalar to $q$ , it may be assumed that

\omega (\lambda )\neq 0~~{\text{ if }}\lambda \notin [1,\infty ).

\omega (\lambda )\neq 0~~{\text{ if }}\lambda \notin [1,\infty ).

If $ω (λ) \neq 0$ , the Green's function $G λ (x, y)$ at $λ$ is defined by

{\displaystyle G_{\lambda }(x,y)={\begin{cases}\Phi _{\lambda }(

G_{\lambda }(x,y)={\begin{cases}\Phi _{\lambda }(x)\mathrm {X} _{\lambda }(y)/\omega (\lambda )&(x\leq y),\\[1ex]\mathrm {X} _{\lambda }(x)\Phi _{\lambda }(y)/\omega (\lambda )&(x\geq y).\end{cases}}

and is independent of the choice of

Φ λ

and

Χ λ

In the examples there will be a third "bad" eigenfunction $Ψ λ$ defined and holomorphic for $λ$ not in $[1, \infty)$ such that $Ψ λ$ satisfies the boundary conditions at neither $0$ nor $\infty$ . This means that for $λ$ not in $[1, \infty)$

$W (Φ λ,Ψ λ)$ is nowhere vanishing;
$W (Χ λ,Ψ λ)$ is nowhere vanishing.

In this case $Χ λ$ is proportional to $Φ λ + m (λ) Ψ λ$ , where

m(\lambda )=-W(\Phi _{\lambda },\mathrm {X} _{\lambda })/W(\Psi _{\lambda },\mathrm {X} _{\lambda }).

m(\lambda )=-W(\Phi _{\lambda },\mathrm {X} _{\lambda })/W(\Psi _{\lambda },\mathrm {X} _{\lambda }).

Let $H 1$ be the space of square integrable continuous functions on $(0, \infty)$ and let $H 0$ be

the space of $C 2$ functions $f$ on $(0, \infty)$ ofcompact supportif $D$ is limit point at $0$
the space of $C 2$ functions $f$ on $(0, \infty)$ with $W (f, Φ 0) = 0$ at $0$ and with $f = 0$ near $\infty$ if $D$ is limit circle at $0$ .

Define $T = G 0$ by

{\displaystyle (

(Tf)(x)=\int _{0}^{\infty }G_{0}(x,y)f(y)\,dy.

Then $T D = I$ on $H 0$ , $D T = I$ on $H 1$ and the operator $D$ is bounded below on $H 0$ :

(Df,f)\geq (f,f).

(Df,f)\geq (f,f).

Thus $T$ is a self-adjoint bounded operator with $0 \leq T \leq I$ .

Formally $T = D -1$ . The corresponding operators $G λ$ defined for $λ$ not in $[1, \infty)$ can be formally identified with

(D-\lambda )^{-1}=T(I-\lambda T)^{-1}

(D-\lambda )^{-1}=T(I-\lambda T)^{-1}

and satisfy

G λ (D - λ) = I

H 0

(D - λ) G λ = I

H 1

Spectral theorem and Titchmarsh–Kodaira formula[edit]

Theorem.^[10]^[14]^[15] — For every real number $λ$ let $ρ (λ)$ be defined by the Titchmarsh–Kodaira formula:

\rho (\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}{\frac {1}{\pi }}\int _{\delta }^{\lambda +\delta }\operatorname {Im} m(t+i\varepsilon )\,dt.

\rho (\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}{\frac {1}{\pi }}\int _{\delta }^{\lambda +\delta }\operatorname {Im} m(t+i\varepsilon )\,dt.

Then $ρ (λ)$ is a lower semicontinuous non-decreasing function of $λ$ and if

{\displaystyle (

(Uf)(\lambda )=\int _{0}^{\infty }f(x)\Phi (x,\lambda )\,dx,

then

U

defines a unitary transformation of

L 2 (0, \infty)

onto

L 2 ([1,\infty), dρ)

such that

UDU -1

corresponds to multiplication by

λ

The inverse transformation $U -1$ is given by

{\displaystyle (U^{-1}g)(

(U^{-1}g)(x)=\int _{1}^{\infty }g(\lambda )\Phi (x,\lambda )\,d\rho (\lambda ).

The spectrum of $D$ equals the support of $dρ$ .

Kodaira gave a streamlined version^[16]^[17] of Weyl's original proof.^[10] (M.H. Stone had previously shown^[18] how part of Weyl's work could be simplified using von Neumann's spectral theorem.)

In fact for $T = D -1$ with $0 \leq T \leq I$ , the spectral projection $E (λ)$ of $T$ is defined by

{\displaystyle E(\lambda )=\chi _{[\lambda ^{-1},1]}(

E(\lambda )=\chi _{[\lambda ^{-1},1]}(T)

It is also the spectral projection of $D$ corresponding to the interval $[1, λ]$ .

For $f$ in $H 1$ define

{\displaystyle f(x,\lambda )=(E(\lambda )f)(

f(x,\lambda )=(E(\lambda )f)(x).

$f (x, λ)$ may be regarded as a differentiable map into the space of functions $ρ$ of bounded variation; or equivalently as a differentiable map

{\displaystyle x\mapsto (d_{\lambda }f)(

x\mapsto (d_{\lambda }f)(x)

into the Banach space

E

of bounded linear functionals

dρ

[C [α, β]]

for any compact subinterval

[α, β]

[1, \infty)

The functionals (or measures) $d λ f (x)$ satisfies the following $E$ -valued second order ordinary differential equation:

D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f,

D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f,

with initial conditions at

c

(0, \infty)

{\displaystyle (d_{\lambda }f)(

(d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot )=\mu ^{(0)},\quad (d_{\lambda }f)^{\prime }(c)=d_{\lambda }f_{x}(c,\cdot )=\mu ^{(1)}.

If $φ λ$ and $χ λ$ are the special eigenfunctions adapted to $c$ , then

{\displaystyle d_{\lambda }f(

d_{\lambda }f(x)=\varphi _{\lambda }(x)\mu ^{(0)}+\chi _{\lambda }(x)\mu ^{(1)}.

Moreover,

{\displaystyle \mu ^{(

\mu ^{(k)}=d_{\lambda }(f,\xi _{\lambda }^{(k)}),

where

{\displaystyle \xi _{\lambda }^{(

\xi _{\lambda }^{(k)}=DE(\lambda )\eta ^{(k)},

with

{\displaystyle \eta _{z}^{(0)}(

\eta _{z}^{(0)}(y)=G_{z}(c,y),\,\,\,\,\eta _{z}^{(1)}(x)=\partial _{x}G_{z}(c,y),\,\,\,\,(z\notin [1,\infty )).

(As the notation suggests,

ξ λ (0)

and

ξ λ (1)

do not depend on the choice of

z

Setting

{\displaystyle \sigma _{ij}(\lambda )=(\xi _{\lambda }^{(

\sigma _{ij}(\lambda )=(\xi _{\lambda }^{(i)},\xi _{\lambda }^{(j)}),

it follows that

{\displaystyle d_{\lambda }(E(\lambda )\eta _{z}^{(

d_{\lambda }(E(\lambda )\eta _{z}^{(i)},\eta _{z}^{(j)})=|\lambda -z|^{-2}\cdot d_{\lambda }\sigma _{ij}(\lambda ).

On the other hand, there are holomorphic functions $a (λ)$ , $b (λ)$ such that

$φ λ + a (λ) χ λ$ is proportional to $Φ λ$ ;
$φ λ + b (λ) χ λ$ is proportional to $Χ λ$ .

Since $W (φ λ, χ λ) = 1$ , the Green's function is given by

{\displaystyle G_{\lambda }(x,y)={\begin{cases}{\dfrac {(\varphi _{\lambda }(

G_{\lambda }(x,y)={\begin{cases}{\dfrac {(\varphi _{\lambda }(x)+a(\lambda )\chi _{\lambda }(x))(\varphi _{\lambda }(y)+b(\lambda )\chi _{\lambda }(y))}{b(\lambda )-a(\lambda )}}&(x\leq y),\\[1ex]{\dfrac {(\varphi _{\lambda }(x)+b(\lambda )\chi _{\lambda }(x))(\varphi _{\lambda }(y)+a(\lambda )\chi _{\lambda }(y))}{b(\lambda )-a(\lambda )}}&(y\leq x).\end{cases}}

Direct calculation^[19] shows that

{\displaystyle (\eta _{z}^{(

(\eta _{z}^{(i)},\eta _{z}^{(j)})=\operatorname {Im} M_{ij}(z)/\operatorname {Im} z,

where the so-called characteristic matrix

M ij (z)

is given by

{\displaystyle M_{00}(

M_{00}(z)={\frac {a(z)b(z)}{a(z)-b(z)}},\,\,M_{01}(z)=M_{10}(z)={\frac {a(z)+b(z)}{2(a(z)-b(z))}},\,\,M_{11}(z)={\frac {1}{a(z)-b(z)}}.

Hence

{\displaystyle \int _{-\infty }^{\infty }(\operatorname {Im} z)\cdot |\lambda -z|^{-2}\,d\sigma _{ij}(\lambda )=\operatorname {Im} M_{ij}(

\int _{-\infty }^{\infty }(\operatorname {Im} z)\cdot |\lambda -z|^{-2}\,d\sigma _{ij}(\lambda )=\operatorname {Im} M_{ij}(z),

which immediately implies

\sigma _{ij}(\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}\int _{\delta }^{\lambda +\delta }\operatorname {Im} M_{ij}(t+i\varepsilon )\,dt.

\sigma _{ij}(\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}\int _{\delta }^{\lambda +\delta }\operatorname {Im} M_{ij}(t+i\varepsilon )\,dt.

(This is a special case of the "Stieltjes inversion formula".)

Setting $ψ λ (0) = φ λ$ and $ψ λ (1) = χ λ$ , it follows that

{\displaystyle (E(\mu )f)(

(E(\mu )f)(x)=\sum _{i.j}\int _{0}^{\mu }\int _{0}^{\infty }\psi _{\lambda }^{(i)}(x)\psi _{\lambda }^{(j)}(y)f(y)\,dy\,d\sigma _{ij}(\lambda )=\int _{0}^{\mu }\int _{0}^{\infty }\Phi _{\lambda }(x)\Phi _{\lambda }(y)f(y)\,dy\,d\rho (\lambda ).

This identity is equivalent to the spectral theorem and Titchmarsh–Kodaira formula.

Application to the hypergeometric equation[edit]

The Mehler–Fock transform^[20]^[21]^[22] concerns the eigenfunction expansion associated with the Legendre differential operator $D$

Df=-((x^{2}-1)f')'=-(x^{2}-1)f''-2xf'

Df=-((x^{2}-1)f')'=-(x^{2}-1)f''-2xf'

(1, \infty)

. The eigenfunctions are the Legendre functions^[23]

P_{-1/2+i{\sqrt {\lambda }}}(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }\left({\sin \theta +ie^{-r}\cos \theta  \over \cos \theta -ie^{-r}\sin \theta }\right)^{{1 \over 2}+i{\sqrt {\lambda }}}\,d\theta

P_{-1/2+i{\sqrt {\lambda }}}(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }\left({\sin \theta +ie^{-r}\cos \theta  \over \cos \theta -ie^{-r}\sin \theta }\right)^{{1 \over 2}+i{\sqrt {\lambda }}}\,d\theta

with eigenvalue

λ \geq 0

. The two Mehler–Fock transformations are^[24]

{\displaystyle Uf(\lambda )=\int _{1}^{\infty }f(

Uf(\lambda )=\int _{1}^{\infty }f(x)\,P_{-1/2+i{\sqrt {\lambda }}}(x)\,dx

and

{\displaystyle U^{-1}g(

U^{-1}g(x)=\int _{0}^{\infty }g(\lambda )\,{1 \over 2}\tanh \pi {\sqrt {\lambda }}\,d\lambda .

(Often this is written in terms of the variable $τ = \sqrt λ$ .)

Mehler and Fock studied this differential operator because it arose as the radial component of the Laplacian on 2-dimensional hyperbolic space. More generally,^[25] consider the group $G = SU(1,1)$ consisting of complex matrices of the form

{\begin{bmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}

{\begin{bmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}

with determinant $| α | 2 - | β | 2 = 1$ .

Application to the hydrogen atom[edit]

Generalisations and alternative approaches[edit]

A Weyl function can be defined at a singular endpoint $a$ giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory.^[26] this applies for example to the case of radial Schrödinger operators

{\displaystyle Df=-f''+{\frac {\ell (\ell +1)}{x^{2}}}f+V(

Df=-f''+{\frac {\ell (\ell +1)}{x^{2}}}f+V(x)f,\qquad x\in (0,\infty )

The whole theory can also be extended to the case where the coefficients are allowed to be measures.^[27]

Gelfand–Levitan theory[edit]

Notes[edit]

^ This is a limit in the strong operator topology.

^ Abona fide inner product is defined on the quotient by the subspace of null functions

f

, i.e. those with

\mu _{\xi }(|f|_{2})=0

. Alternatively in this case the support of the measure is

{\displaystyle \sigma (

, so the right hand side defines a (non-degenerate) inner product on

{\displaystyle C(\sigma (

References[edit]

Citations[edit]

^ Titchmarsh 1962, p. 22

^ Dieudonné 1969, Chapter X

^ Courant & Hilbert 1989

^ Titchmarsh 1962

^ Titchmarsh 1939, §8.2

^ Burkill 1951, pp. 50–52

^ Loomis 1953, p. page 40

^ Loomis 1953, pp. 30–31

^ Kolmogorov & Fomin 1975, pp. 374–376

^ ^a ^b ^c ^d ^e Weyl 1910.^[specify]

^ ^a ^b Bellman 1969, p. 116

^ Reed & Simon 1975, p. 159

^ Reed & Simon 1975, p. 154

^ Titchmarsh 1946, Chapter III

^ Kodaira 1949, pp. 935–936

^ Kodaira 1949, pp. 929–932; for omitted details, see Kodaira 1950, pp. 529–536

^ Dieudonné 1988

^ Stone 1932, Chapter X

^ Kodaira 1950, pp. 534–535

^ Mehler 1881.

^ Fock 1943, pp. 253–256

^ Vilenkin 1968

^ Terras 1984, pp. 261–276

^ Lebedev 1972

^ Vilenkin 1968, Chapter VI

^ Kostenko, Sakhnovich & Teschl 2012, pp. 1699–1747

^ Eckhardt & Teschl 2013, pp. 151–224

Bibliography[edit]

Akhiezer, Naum Ilich; Glazman, Izrael Markovich (1993), Theory of Linear Operators in Hilbert Space, Dover, ISBN 978-0-486-67748-4

Bellman, Richard (1969), Stability Theory of Differential Equations, Dover, ISBN 978-0-486-62210-1

Burkill, J.C. (1951), The Lebesgue Integral, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 40, Cambridge University Press, ISBN 978-0-521-04382-3

Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential equations, McGraw-Hill, ISBN 978-0-07-011542-2

Courant, Richard; Hilbert, David (1989), Method of Mathematical Physics, Vol. I, Wiley-Interscience, ISBN 978-0-471-50447-4

Dieudonné, Jean (1969), Treatise on Analysis, Vol. I [Foundations of Modern Analysis], Academic Press, ISBN 978-1-4067-2791-3

Dieudonné, Jean (1988), Treatise on Analysis, Vol. VIII, Academic Press, ISBN 978-0-12-215507-9

Dunford, Nelson; Schwartz, Jacob T. (1963), Linear Operators, Part II Spectral Theory. Self Adjoint Operators in Hilbert space, Wiley Interscience, ISBN 978-0-471-60847-9

Fock, V.A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. Acad. Sci. URSS, 39

Hille, Einar (1969), Lectures on Ordinary Differential Equations, Addison-Wesley, ISBN 978-0-201-53083-4

Kodaira, Kunihiko (1949), "The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices", American Journal of Mathematics, 71 (4): 921–945, doi:10.2307/2372377, JSTOR 2372377

Kodaira, Kunihiko (1950), "On ordinary differential equations of any even order and the corresponding eigenfunction expansions", American Journal of Mathematics, 72 (3): 502–544, doi:10.2307/2372051, JSTOR 2372051

Kolmogorov, A.N.; Fomin, S.V. (1975). Introductory Real Analysis. Dover. ISBN 978-0-486-61226-3.

Kostenko, Aleksey; Sakhnovich, Alexander; Teschl, Gerald (2012), "Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials", Int Math Res Notices, 2012, arXiv:1007.0136, doi:10.1093/imrn/rnr065

Lebedev, N.N. (1972), Special Functions and Their Applications, Dover, ISBN 978-0-486-60624-8

Loomis, Lynn H. (1953), An Introduction to Abstract Harmonic Analysis, van Nostrand

Mehler, F.G. (1881), "Ueber mit der Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsverteilung", Mathematische Annalen, 18 (2): 161–194, doi:10.1007/BF01445847, S2CID 122590188

Reed, Michael; Simon, Barry (1975), Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, ISBN 978-0-12-585002-5

Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional Analysis. Dover Publications. ISBN 0-486-66289-6.

Stone, Marshall Harvey (1932), Linear transformations in Hilbert space and Their Applications to Analysis, AMS Colloquium Publications, vol. 16, ISBN 978-0-8218-1015-6

Terras, Audrey (1984), "Non-Euclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities", J. Multivariate Anal., 15 (2), doi:10.1016/0047-259X(84)90031-9

Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. AMS Graduate Studies in Mathematics. Vol. 99. ISBN 978-0-8218-4660-5.

Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. AMS Graduate Studies in Mathematics. Vol. 140. ISBN 978-0-8218-8328-0.

Titchmarsh, Edward Charles (1939), Theory of Functions, Oxford University Press

Titchmarsh, Edward Charles (1946), Eigenfunction expansions associated with second order differential equations, Vol. I (1st ed.), Oxford University Press

Titchmarsh, Edward Charles (1962), Eigenfunction expansions associated with second order differential equations, Vol. I (2nd ed.), Oxford University Press, ISBN 978-0-608-08254-7

Eckhardt, Jonathan; Teschl, Gerald (2013), "Sturm–Liouville operators with measure-valued coefficients", Journal d'Analyse Mathématique, 120, arXiv:1105.3755, doi:10.1007/s11854-013-0018-x

Vilenkin, Naoum Iakovlevitch (1968). Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs. Vol. 22. American Mathematical Society. ISBN 978-0-8218-1572-4.

Weidmann, Joachim (1987). Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics. Vol. 1258. Springer-Verlag. ISBN 978-0-387-17902-5.

Weyl, Hermann (1910a), "Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen", Mathematische Annalen, 68 (2): 220–269, doi:10.1007/BF01474161, S2CID 119727984

Weyl, Hermann (1910b), "Über gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunktionen", Nachr. Akad. Wiss. Göttingen. Math.-Phys.: 442–446

Weyl, Hermann (1935), "Über das Pick-Nevanlinnasche Interpolationsproblem und sein infinitesimales Analogen", Annals of Mathematics, 36 (1): 230–254, doi:10.2307/1968677, JSTOR 1968677

Functional analysis (topics – glossary)

Spaces

Properties

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

Retrieved from "https://en.wikipedia.org/w/index.php?title=Spectral_theory_of_ordinary_differential_equations&oldid=1189800472" Categories: ●Ordinary differential equations ●Operator theory ●Spectral theory Hidden categories: ●Harv and Sfn no-target errors ●Articles needing more detailed references ●Articles with short description ●Short description matches Wikidata ●This page was last edited on 14 December 2023, at 02:43 (UTC). ●Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. ●Privacy policy ●About Wikipedia ●Disclaimers ●Contact Wikipedia ●Code of Conduct ●Developers ●Statistics ●Cookie statement ●Mobile view