Spectral theory of ordinary differential equations#Spectral measure

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) of Harish 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 matrix in quantum mechanics.

{{Main|Ordinary differential equations}}

Let {{math|D}} be the second order differential operator on {{open-open|a, b}} given by

$$Df(x)\; =\; -p(x)\; f\text{'}\text{'}(x)\; +\; r(x)\; f\text{'}(x)\; +\; q(x)\; f(x),$$

where {{math|p}} is a strictly positive continuously differentiable function and {{math|q}} and {{math|r}} are continuous real-valued functions.

For {{math|x₀}} in {{open-open|a, b}}, define the Liouville transformation {{math|ψ}} by

$$\backslash psi(x)\; =\; \backslash int\_\{x\_0\}^x\; p(t)^\{-1/2\}\backslash ,\; dt$$

If

$$U:\; L^2(a,\; b)\; \backslash mapsto\; L^2(\backslash psi(a),\; \backslash psi(b))$$

is the unitary operator defined by

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

then

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

and

$$\backslash begin\{align\}$$

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 12 g \frac{\psi}{\psi'} \right] \cdot \psi' + \frac 1 2 \left[ g' \psi' + \frac 12 g \frac{\psi}{\psi'} \right] \cdot \frac{\psi''}{\psi'} \\[1ex]

& = g \psi'^2 + 2 g' \psi + \frac 12 g \cdot \left[ \frac{\psi'}{\psi'} - \frac 1 2 \frac {\psi^2}{\psi'^2} \right]

\end{align}

Hence,

$$UDU^\{-1\}\; g=\; -g\text{'}\text{'}\; +\; R\; g\text{'}\; +\; Q\; g,$$

where

$$R\; =\; \backslash frac\; \{\; p\text{'}\; +\; r\}\{p^\{1/2\}\}$$

and

$$Q\; =\; q\; -\; \backslash frac\; \{r\; p\text{'}\}\{4p\}\; +\; \backslash frac\; \{p\text{'}\text{'}\}4\; -\; \backslash frac\; \{5\; p\text{'}^2\}\{16\; p\}$$

The term in {{math|g′}} can be removed using an Euler integrating factor. If {{math|1=S′/S = R/2}}, then {{math|1=h = Sg}} satisfies

$$(S\; U\; D\; U^\{-1\}\; S^\{-1\})\; h\; =\; -h\text{'}\text{'}\; +\; V\; h,$$

where the potential V is given by

$$V\; =\; Q\; +\; \backslash frac\{S\text{'}\text{'}\}\{S\}$$

The differential operator can thus always be reduced to one of the form {{harvnb|Titchmarsh|1962|p= 22}}

$$Df\; =\; -\; f\text{'}\text{'}\; +\; qf.$$

The following is a version of the classical Picard existence theorem for second order differential equations with values in a Banach space {{math|E}}.{{harvnb|Dieudonné |1969 | at= Chapter X}}

Let {{math|α}}, {{math|β}} be arbitrary elements of {{math|E}}, {{math|A}} a bounded operator on {{math|E}} and {{math|q}} a continuous function on {{closed-closed|a, b}}.

Then, for {{math|1=c = a}} or {{math|1=c = b}}, the differential equation

$$D\; f\; =\; A\; f$$

has a unique solution {{math|f}} in {{math|C²([a,b], E)}} satisfying the initial conditions

$$f(c)\; =\; \backslash beta\; \backslash ,\; ,\; \backslash ;\; f\text{'}(c)\; =\; \backslash alpha.$$

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

of the integral equation

$$f\; =\; h\; +\; T\; f$$

with {{math|T}} the bounded linear map on {{math|C([a,b], E)}} defined by

$$Tf(x)\; =\; \backslash int\_c^x\; K(x,y)f(y)\; \backslash ,\; dy,$$

where {{math|K}} is the Volterra kernel

$$K(x,\; t)\; =\; (x\; -\; t)\; (\; q(t)\; -\; A)$$

and

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

Since {{math|{{norm|T^k}}}} tends to 0, this integral equation has a unique solution given by the Neumann series

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

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

If {{math|f}} is twice continuously differentiable (i.e. {{math|C²}}) on {{open-open|a, b}} satisfying {{math|1=Df = λf}}, then {{math|f}} is called an eigenfunction of {{math|D}} with eigenvalue {{mvar|λ}}.

• In the case of a compact interval {{closed-closed|a, b}} and {{math|q}} continuous on {{closed-closed|a, b}}, the existence theorem implies that for {{math|1=c = a}} or {{math|1=c = b}} and every complex number {{mvar|λ}} there a unique {{math|C²}} eigenfunction {{math|f_λ}} on {{closed-closed|a, b}} with {{math|f_λ(c)}} and {{math|f′_λ(c)}} prescribed. Moreover, for each {{mvar|x}} in {{closed-closed|a, b}}, {{math|f_λ(x)}} and {{math|f′_λ(x)}} are holomorphic functions of {{mvar|λ}}.
• For an arbitrary interval {{open-open|a, b}} and {{math|q}} continuous on {{open-open|a, b}}, the existence theorem implies that for {{math|c}} in {{open-open|a, b}} and every complex number {{mvar|λ}} there a unique {{math|C²}} eigenfunction {{math|f_λ}} on {{open-open|a, b}} with {{math|f_λ(c)}} and {{math|f′_λ(c)}} prescribed. Moreover, for each {{mvar|x}} in {{open-open|a, b}}, {{math|f_λ(x)}} and {{math|f′_λ(x)}} are holomorphic functions of {{mvar|λ}}.

If {{math|f}} and {{math|g}} are {{math|C²}} functions on {{open-open|a, b}}, the Wronskian {{math|W(f, g)}} is defined by

$$W(f,\; g)\; (x)\; =\; f(x)\; g\text{'}(x)\; -\; f\text{'}(x)\; g(x).$$

Green's formula - which in this one-dimensional case is a simple integration by parts - states that for {{mvar|x}}, {{mvar|y}} in {{open-open|a, b}}

$$\backslash int\_x^y\; (Df)\; g\; -\; f\; (Dg)\; \backslash ,\; dt\; =\; W(f,g)(y)\; -\; W(f,g)(x).$$

When {{math|q}} is continuous and {{math|f}}, {{math|g}} are {{math|C²}} on the compact interval {{closed-closed|a, b}}, this formula also holds for {{math|1=x = a}} or {{math|1=y = b}}.

When {{math|f}} and {{math|g}} are eigenfunctions for the same eigenvalue, then

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

so that {{math|W(f, g)}} is independent of {{mvar|x}}.

{{Main|Sturm–Liouville theory}}

Let {{closed-closed|a, b}} be a finite closed interval, {{math|q}} a real-valued continuous function on {{closed-closed|a, b}} and let {{math|H₀}} be the space of {{math|C²}} functions {{math|f}} on {{closed-closed|a, b}} satisfying the Robin boundary conditions

$$\backslash begin\{cases\}$$

\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

$$(f,g)\; =\; \backslash int\_a^b\; f(x)\; \backslash overline\{g(x)\}\; \backslash ,\; dx.$$

In practice usually one of the two standard boundary conditions:

• Dirichlet boundary condition {{math|1=f(c) = 0}}
• Neumann boundary condition {{math|1=f′(c) = 0}}

is imposed at each endpoint {{math|1=c = a, b}}.

The differential operator {{math|D}} given by

$$Df\; =\; -f\text{'}\text{'}\; +\; qf$$

acts on {{math|H₀}}. A function {{math|f}} in {{math|H₀}} is called an eigenfunction of {{math|D}} (for the above choice of boundary values) if {{math|1=Df = λ f}} for some complex number {{mvar|λ}}, the corresponding eigenvalue. By Green's formula, {{math|D}} is formally self-adjoint on {{math|H₀}}, since the Wronskian {{math|W(f, g)}} vanishes if both {{math|f}}, {{math|g}} satisfy the boundary conditions:

$$(Df,\; g)\; =\; (f,\; Dg),\; \backslash quad\; \backslash text\{\; for\; \}\; f,\; g\; \backslash in\; H\_0.$$

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

• the eigenvalues of {{math|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{{harvnb|Courant | Hilbert |1989}} (see below). In fact it is easy to see a priori that the eigenvalues are bounded below because the operator {{math|D}} is itself bounded below on {{math|H₀}}:

{{block indent | text = $(Df,\; f)\; \backslash ge\; M\; (f,\; f)$ for some finite (possibly negative) constant $M$.}}

In fact, integrating by parts,

$$(Df,f)\; =\; \backslash left[-f\text{'}\; \backslash overline\{f\}\backslash right]\_a^b\; +\; \backslash int\; |f\text{'}|^2\; +\; \backslash int\; q\; |f|^2.$$

For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with {{math|1=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 {{math|ε > 0}}, there is constant {{math|R > 0}} such that {{math|{{abs|f(x)}}² ≤ ε (f′, f′) + R (f, f)}} for all {{math|f}} in {{math|C¹[a, b]}}."

In fact, since

$$|f(b)\; -\; f(x)|\; \backslash leq\; (b-a)^\{1/2\}\; \backslash cdot\; \backslash |\; f\text{'}\; \backslash |\_2,$$

only an estimate for {{math|f(b)}} is needed and this follows by replacing {{math|f(x)}} in the above inequality by {{math|(x − a)ⁿ·(b − a)⁻ⁿ·f(x)}} for {{mvar|n}} sufficiently large.

From the theory of ordinary differential equations, there are unique fundamental eigenfunctions {{math|φ_λ(x)}}, {{math|χ_λ(x)}} such that

• {{math|1=D φ_λ = λ φ_λ}}, {{math|1=φ_λ(a) = sin α}}, {{math|1=φ_λ'(a) = cos α}}
• {{math|1=D χ_λ = λ χ_λ}}, {{math|1=χ_λ(b) = sin β}}, {{math|1=χ_λ'(b) = cos β}}

which at each point, together with their first derivatives, depend holomorphically on {{mvar|λ}}. Let

$$\backslash omega(\backslash lambda)\; =\; W(\backslash phi\_\backslash lambda,\; \backslash chi\_\backslash lambda),$$

be an entire holomorphic function.

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

If {{mvar|λ}} is not an eigenvalue of {{math|D}} on {{math|H₀}}, define the Green's function by

$$G\_\backslash lambda(x,\; y)\; =\; \backslash begin\{cases\}$$

\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 {{math|C[a,b]}} via

$$(G\_\backslash lambda\; f)(x)\; =\backslash int\_a^b\; G\_\backslash lambda(x,y)\; f(y)\backslash ,\; dy.$$

Since {{math|G_λ(x,y)}} is continuous on {{math|[a, b] × [a, b]}}, it defines a Hilbert–Schmidt operator on the Hilbert space completion {{math|H}} of {{math|1=C[a, b] = H₁}} (or equivalently of the dense subspace {{math|H₀}}), taking values in {{math|H₁}}. This operator carries {{math|H₁}} into {{math|H₀}}. When {{mvar|λ}} is real, {{math|1=G_λ(x,y) = G_λ(y,x)}} is also real, so defines a self-adjoint operator on {{math|H}}. Moreover,

• {{math|1=G_λ (D − λ) = I}} on {{math|H₀}}
• {{math|G_λ}} carries {{math|H₁}} into {{math|H₀}}, and {{math|1=(D − λ) G_λ = I}} on {{math|H₁}}.

Thus the operator {{math|G_λ}} can be identified with the resolvent {{math|(D − λ)⁻¹}}.

{{math theorem | math_statement = The eigenvalues of D are real of multiplicity one and form an increasing sequence {{math|λ₁ < λ₂ < ⋯}} tending to infinity.

The corresponding normalised eigenfunctions form an orthonormal basis of {{math|H₀}}.

The {{mvar|k}}-th eigenvalue of {{math|D}} is given by the minimax principle

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

In particular if {{math|q₁ ≤ q₂}}, then

$$\backslash lambda\_k(D\_1)\; \backslash le\; \backslash lambda\_k(D\_2).$$}}

In fact let {{math|1=T = G_λ}} for {{mvar|λ}} large and negative. Then {{math|T}} defines a compact self-adjoint operator on the Hilbert space {{math|H}}.

By the spectral theorem for compact self-adjoint operators, {{math|H}} has an orthonormal basis consisting of eigenvectors {{math|ψ_n}} of {{math|T}} with {{math|1=Tψ_n = μ_n ψ_n}}, where {{math|μ_n}} tends to zero. The range of {{math|T}} contains {{math|H₀}} so is dense. Hence 0 is not an eigenvalue of {{math|T}}. The resolvent properties of {{math|T}} imply that {{math|ψ_n}} lies in {{math|H₀}} and that

$$D\; \backslash psi\_n\; =\; \backslash left(\backslash lambda\; +\; \backslash frac\{1\}\{\backslash mu\_n\}\backslash right)\; \backslash psi\_n$$

The minimax principle follows because if

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

then {{math|1=λ(G) = λ_k}} for the linear span of the first {{math|k − 1}} eigenfunctions. For any other {{math|(k − 1)}}-dimensional subspace {{math|G}}, some {{math|f}} in the linear span of the first {{mvar|k}} eigenvectors must be orthogonal to {{math|G}}. Hence {{math|λ(G) ≤ (Df,f)/(f,f) ≤ λ_k}}.

For simplicity, suppose that {{math|m ≤ q(x) ≤ M}} on {{closed-closed|0, π}} with Dirichlet boundary conditions. The minimax principle shows that

$$n^2\; +\; m\; \backslash le\; \backslash lambda\_n(D)\; \backslash le\; n^2\; +\; M.$$

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

The Dirichlet boundary conditions imply that

$$\backslash omega(\backslash lambda)\; =\; \backslash phi\_\backslash lambda(b).$$

Using Picard iteration, Titchmarsh showed that {{math|φ_λ(b)}}, and hence {{math|ω(λ)}}, is an entire function of finite order {{math|1/2}}:

$$\backslash omega(\backslash lambda)\; =\; \backslash mathcal\; O\backslash left(e^\{\backslash sqrt$$

At a zero {{math|μ}} of {{math|ω(λ)}}, {{math|1=φ_μ(b) = 0}}. Moreover,

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

satisfies {{math|1=(D − μ)ψ = φ_μ}}. Thus

$$\backslash omega(\backslash lambda)\; =\; (\backslash lambda\; -\; \backslash mu)\; \backslash psi(b)\; +\; \backslash mathcal\; O(\; (\backslash lambda\; -\; \backslash mu)^2)$$

This implies that{{harvnb|Titchmarsh|1962}}

{{block indent | text = {{math|μ}} is a simple zero of {{math|ω(λ)}}.}}

For otherwise {{math|1=ψ(b) = 0}}, so that {{math|ψ}} would have to lie in {{math|H₀}}. But then

$$(\backslash phi\_\backslash mu,\; \backslash phi\_\backslash mu)\; =\; ((D\; -\; \backslash mu)\backslash psi,\; \backslash phi\_\backslash mu)\; =\; (\backslash psi,\; (D\; -\; \backslash mu)\; \backslash phi\_\backslash mu)\; =\; 0,$$

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{{harvnb|Titchmarsh|1939|at=§8.2}}

$$\backslash omega(\backslash lambda)\; =\; C\; \backslash prod\; (1\; -\backslash lambda/\backslash lambda\_n),$$

for some non-zero constant {{math|C}}.

Hence

$$\backslash det\; (\; I\; -\; \backslash mu(D\; -\; \backslash lambda)^\{-1\})\; =\; \backslash prod\; \backslash left(\; 1\; -\; \{\backslash mu\; \backslash over\; \backslash lambda\_n\; -\backslash lambda\}\backslash right)\; =\; \backslash prod\; \{1\; -\; (\backslash lambda+\backslash mu)/\backslash lambda\_n\; \backslash over\; 1-\backslash lambda/\backslash lambda\_n\}\; =\; \{\backslash omega(\backslash lambda+\backslash mu)\backslash over\; \backslash omega(\backslash lambda)\}\; .$$

In particular if 0 is not an eigenvalue of {{math|D}}

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

{{See also|Bounded variation|Lebesgue–Stieltjes integration|Riesz representation theorem}}

A function {{math|ρ(x)}} of bounded variation{{harvnb|Burkill|1951|pp=50–52}} on a closed interval {{closed-closed|a, b}} is a complex-valued function such that its total variation {{math|V(ρ)}}, the supremum of the variations

$$\backslash sum\_\{r=0\}^\{k-1\}\; |\backslash rho(x\_\{r+1\})\; -\; \backslash rho(x\_r)|$$

over all dissections

is finite. The real and imaginary parts of {{math|ρ}} are real-valued functions of bounded variation. If {{math|ρ}} is real-valued and normalised so that {{math|1=ρ(a) = 0}}, it has a canonical decomposition as the difference of two bounded non-decreasing functions:

$$\backslash rho(x)\; =\; \backslash rho\_+(x)\; -\; \backslash rho\_-(x),$$

where {{math|ρ₊(x)}} and {{math|ρ_–(x)}} are the total positive and negative variation of {{math|ρ}} over {{closed-closed|a, x}}.

If {{math|f}} is a continuous function on {{closed-closed|a, b}} its Riemann–Stieltjes integral with respect to {{math|ρ}}

$$\backslash int\_a^b\; f(x)\backslash ,\; d\backslash rho(x)$$

is defined to be the limit of approximating sums

$$\backslash sum\_\{r=0\}^\{k-1\}\; f(x\_r)(\backslash rho(x\_\{r+1\})-\backslash rho(x\_r))$$

as the mesh of the dissection, given by {{math|sup {{abs|x_r+1 − x_r}}}}, tends to zero.

This integral satisfies

$$\backslash left|\backslash int\_a^b\; f(x)\backslash ,\; d\backslash rho(x)\backslash right|\backslash le\; V(\backslash rho)\backslash cdot\; \backslash |f\backslash |\_\backslash infty$$

and thus defines a bounded linear functional {{math|dρ}} on {{math|C[a, b]}} with norm {{math|1={{norm|dρ}} = V(ρ)}}.

Every bounded linear functional {{math|μ}} on {{math|C[a, b]}} has an absolute value |μ| defined for non-negative {{math|f}} by{{harvnb|Loomis|1953|p=[https://archive.org/details/introductiontoab031610mbp page 40]}}

$$|\backslash mu|(f)\; =\; \backslash sup\_\{0\backslash le\; |g|\backslash le\; f\}\; |\backslash mu(g)|.$$

The form {{math|{{abs|μ}}}} extends linearly to a bounded linear form on {{math|C[a, b]}} with norm {{math|{{norm|μ}}}} and satisfies the characterizing inequality

$$|\backslash mu(f)|\; \backslash leq\; |\backslash mu|\; (|f|)$$

for {{math|f}} in {{math|C[a, b]}}. If {{math|μ}} is real, i.e. is real-valued on real-valued functions, then

$$\backslash mu\; =\; |\backslash mu|\; -\; (|\backslash mu|\; -\backslash mu)\backslash equiv\; \backslash mu\_+-\backslash mu\_-$$

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

Every positive form {{math|μ}} extends uniquely to the linear span of non-negative bounded lower semicontinuous functions {{math|g}} by the formula{{harvnb|Loomis|1953|pp=30–31}}

$$\backslash mu(g)\; =\; \backslash lim\; \backslash mu(f\_n),$$

where the non-negative continuous functions {{math|f_n}} increase pointwise to {{math|g}}.

The same therefore applies to an arbitrary bounded linear form {{math|μ}}, so that a function {{math|ρ}} of bounded variation may be defined by{{harvnb|Kolmogorov|Fomin|1975|pp=[https://archive.org/details/introductoryreal00kolm_0/page/374 374–376]}}

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

where {{math|χ_A}} denotes the characteristic function of a subset {{math|A}} of {{closed-closed|a, b}}. Thus {{math|1=μ = dρ}} and {{math|1={{norm|μ}} = {{norm|dρ}}}}.

Moreover {{math|1=μ₊ = dρ₊}} and {{math|1=μ_– = dρ_–}}.

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

The support of {{math|1=μ = dρ}} is the complement of all points {{mvar|x}} in {{closed-closed|a, b}} where {{math|ρ}} is constant on some neighborhood of {{mvar|x}}; by definition it is a closed subset {{math|A}} of {{closed-closed|a, b}}. Moreover, {{math|1=μ((1 − χ_A)f) = 0}}, so that {{math|1=μ(f) = 0}} if {{math|f}} vanishes on {{math|A}}.

{{See also|Spectral theorem|Projection-valued measure}}

Let {{math|H}} be a Hilbert space and $T$ a self-adjoint bounded operator on {{math|H}} with $0\; \backslash leq\; T\; \backslash leq\; I$, so that the spectrum $\backslash sigma(T)$ of $T$ is contained in $[0,1]$. If $p(t)$ is a complex polynomial, then by the spectral mapping theorem

$$\backslash sigma\; (p(T))\; =\; p\; (\backslash sigma(T))$$

and hence

$$\backslash |p(T)\backslash |\; \backslash leq\; \backslash |p\backslash |\_\backslash infty$$

where $\backslash |\; \backslash cdot\; \backslash |\_\backslash infty$ denotes the uniform norm on {{math|C[0, 1]}}. By the Weierstrass approximation theorem, polynomials are uniformly dense in {{math|C[0, 1]}}. It follows that $f(T)$ can be defined $\backslash forall\; f\; \backslash in\; C[0,1]$, with

$$\backslash sigma\; (f(T))\; =\; f(\backslash sigma(T))$$ and $$\backslash |\; f(T)\; \backslash |\; \backslash leq\; \backslash |f\backslash |\_\backslash infty.$$

If $0\; \backslash leq\; g\; \backslash leq\; 1$ is a lower semicontinuous function on {{closed-closed|0, 1}}, for example the characteristic function $\backslash chi\_\{[0,\backslash alpha]\}$ of a subinterval of {{closed-closed|0, 1}}, then

$g$ is a pointwise increasing limit of non-negative $f\_n\; \backslash in\; C[0,1]$.

If $\backslash xi$ is a vector in {{math|H}}, then the vectors

$$\backslash eta\_n\; =\; f\_n(T)\backslash xi$$

form a Cauchy sequence in {{math|H}}, since, for $n\; \backslash geq\; m$,

$$\backslash |\backslash eta\_n-\backslash eta\_m\backslash |^2\; \backslash le\; (\backslash eta\_n,\backslash xi)\; -\; (\backslash eta\_m,\backslash xi),$$

and $(\; \backslash eta\_n,\; \backslash xi)\; =\; (f\_n(T)\backslash xi,\; \backslash xi)$ is bounded and increasing, so has a limit.

It follows that $g(T)$ can be defined by{{efn|1=This is a limit in the strong operator topology.}}

$$g(T)\backslash xi\; =\; \backslash lim\; f\_n(T)\backslash xi.$$

If $\backslash xi$ and {{math|η}} are vectors in {{math|H}}, then

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

defines a bounded linear form $\backslash mu\_\{\backslash xi,\; \backslash eta\}$ on {{math|H}}. By the Riesz representation theorem

$$\backslash mu\_\{\backslash xi,\backslash eta\}\; =\; d\backslash rho\_\{\backslash xi,\backslash eta\}$$

for a unique normalised function $\backslash rho\_\{\backslash xi,\; \backslash eta\}$ of bounded variation on {{closed-closed|0, 1}}.

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

The operator $g(T)$ is accordingly uniquely characterised by the equation

$$(g(T)\backslash xi,\backslash eta)\; =\; \backslash mu\_\{\backslash xi,\backslash eta\}(g)\; =\; \backslash int\_0^1\; g(\backslash lambda)\; \backslash ,\; d\backslash rho\_\{\backslash xi,\backslash eta\}(\backslash lambda).$$

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

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

so that

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

It follows that

$$g(T)\; =\; \backslash int\_0^1\; g(\backslash lambda)\; \backslash ,\; dE(\backslash lambda),$$

which is understood in the sense that for any vectors $\backslash xi$ and $\backslash eta$,

$$(g(T)\backslash xi,\backslash eta)\; =\; \backslash int\_0^1\; g(\backslash lambda)\backslash ,\; d(E(\backslash lambda)\backslash xi,\backslash eta)\; =\; \backslash int\_0^1\; g(\backslash lambda)\backslash ,\; d\backslash rho\_\{\backslash xi,\backslash eta\}\; (\backslash lambda).$$

For a single vector $\backslash xi,\; \backslash ,\; \backslash mu\_\{\backslash xi\}\; =\; \backslash mu\_\{\backslash xi,\; \backslash xi\}$ is a positive form on {{closed-closed|0, 1}}

(in other words proportional to a probability measure on {{closed-closed|0, 1}}) and $\backslash rho\_\{\backslash xi\}\; =\; \backslash rho\_\{\backslash xi,\; \backslash xi\}$ is non-negative and non-decreasing. Polarisation shows that all the forms $\backslash mu\_\{\backslash xi,\; \backslash eta\}$ can naturally be expressed in terms of such positive forms, since

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

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

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

satisfies

$$(Uf\_1,Uf\_2)=\; \backslash int\_0^1\; f\_1(\backslash lambda)\; \backslash overline\{f\_2(\backslash lambda)\}\; \backslash ,\; d\backslash rho\_\backslash xi(\backslash lambda).$$

Let $L\_2([0,1],\; d\backslash rho\_\backslash xi)$ denote the Hilbert space completion of $C[0,1]$ associated with the possibly degenerate inner product on the right hand side.{{efn|1=A bona fide inner product is defined on the quotient by the subspace of null functions $f$, i.e. those with $\backslash mu\_\backslash xi(|f|\_2)\; =\; 0$. Alternatively in this case the support of the measure is $\backslash sigma(T)$, so the right hand side defines a (non-degenerate) inner product on $C(\backslash sigma(T))$.}}

Thus $U$ extends to a unitary transformation of $L\_2([0,1],\; \backslash rho\_\backslash xi)$ onto {{math|H}}. $U\; T\; U^\backslash ast$ is then just multiplication by $\backslash lambda$ on $L\_2([0,1],\; d\backslash rho\_\backslash xi)$; and more generally $U\; f(T)\; U^\backslash ast$ is multiplication by $f(\backslash lambda)$. In this case, the support of $d\backslash rho\_\backslash xi$ is exactly $\backslash sigma(T)$, so that

{{block indent | text = the self-adjoint operator becomes a multiplication operator on the space of functions on its spectrum with inner product given by the spectral measure.}}

The eigenfunction expansion associated with singular differential operators of the form

$$Df\; =\; -(pf\text{'})\text{'}\; +\; qf$$

on an open interval {{open-open|a, b}} requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints {{math|a}} and {{math|b}} to determine possible boundary conditions there. Unlike the regular Sturm–Liouville case, in some circumstances spectral values of {{math|D}} can have multiplicity 2. In the development outlined below standard assumptions will be imposed on {{math|p}} and {{math|q}} that guarantee that the spectrum of {{math|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 {{math|D}}, {{math|(D + R)⁻¹}} for {{math|R}} large and positive, is given by an operator {{math|T}} corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case {{math|T}} was a compact self-adjoint operator; in this case {{math|T}} is just a self-adjoint bounded operator with {{math|0 ≤ T ≤ I}}. The abstract theory of spectral measure can therefore be applied to {{math|T}} to give the eigenfunction expansion for {{math|D}}.

The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of {{math|D}} lies in {{closed-open|1, ∞}} and that {{math|1=T = D⁻¹}} and let

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

be the spectral projection of {{math|D}} corresponding to the interval {{closed-closed|1, λ}}. For an arbitrary function {{math|f}} define

$$f(x,\backslash lambda)\; =\; (E(\backslash lambda)f)(x).$$

{{math|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

$$x\backslash mapsto\; (d\_\backslash lambda\; f)(x)$$

into the Banach space {{math|E}} of bounded linear functionals {{math|dρ}} on {{math|C[α,β]}} whenever {{closed-closed|α, β}} is a compact subinterval of {{closed-open|1, ∞}}.

Weyl's fundamental observation was that {{math|d_λ f}} satisfies a second order ordinary differential equation taking values in {{math|E}}:

$$D\; (d\_\backslash lambda\; f)\; =\; \backslash lambda\; \backslash cdot\; d\_\backslash lambda\; f.$$

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

$$(d\_\backslash lambda\; f)(c)\; =\; d\_\backslash lambda\; f(c,\backslash cdot),\; \backslash quad\; (d\_\backslash lambda\; f)^\backslash prime(c)\; =\; d\_\backslash lambda\; f\_x(c,\backslash cdot).$$

This point of view may now be turned on its head: {{math|f(c, λ)}} and {{math|f_x(c, λ)}} may be written as

$$f(c,\backslash lambda)\; =\; (f,\backslash xi\_1(\backslash lambda)),\; \backslash quad\; f\_x(c,\backslash lambda)\; =\; (f,\backslash xi\_2(\backslash lambda)),$$

where {{math|ξ₁(λ)}} and {{math|ξ₂(λ)}} are given purely in terms of the fundamental eigenfunctions.

The functions of bounded variation

$$\backslash sigma\_\{ij\}(\backslash lambda)\; =\; (\backslash xi\_i(\backslash lambda),\backslash xi\_j(\backslash lambda))$$

determine a spectral measure on the spectrum of {{math|D}} and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the Titchmarsh–Kodaira formula).

Let {{math|q(x)}} be a continuous real-valued function on {{open-open|0, ∞}} and let {{math|D}} be the second order differential operator

$$Df\; =\; -f\text{'}\text{'}\; +\; qf$$

on {{open-open|0, ∞}}. Fix a point {{math|c}} in {{open-open|0, ∞}} and, for complex {{mvar|λ}}, let $\backslash varphi\_\backslash lambda,\; \backslash theta\_\backslash lambda$ be the unique fundamental eigenfunctions of {{math|D}} on {{open-open|0, ∞}} satisfying

$$(D-\backslash lambda)\backslash varphi\_\backslash lambda\; =\; 0,\; \backslash quad\; (D-\backslash lambda)\backslash theta\_\backslash lambda\; =0$$

together with the initial conditions at {{math|c}}

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

Then their Wronskian satisfies

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

since it is constant and equal to 1 at {{math|c}}.

Let {{mvar|λ}} be non-real and {{math|0 < x < ∞}}. If the complex number $\backslash mu$ is such that $f\; =\; \backslash varphi\; +\; \backslash mu\; \backslash theta$ satisfies the boundary condition $\backslash cos\backslash beta\backslash ,\; f(x)\; -\; \backslash sin\backslash beta\backslash ,\; f\text{'}(x)\; =\; 0$ for some $\backslash beta$ (or, equivalently, $f\text{'}(x)\; /\; f(x)$ is real) then, using integration by parts, one obtains

$$\backslash operatorname\{Im\}(\backslash lambda)\; \backslash int\_c^x\; |\backslash varphi\; +\backslash mu\; \backslash theta|^2\; =\; \backslash operatorname\{Im\}(\backslash mu).$$

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

if {{math|x > c}} and by

if {{math|x < c}}.

Let {{math|D_x}} be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as {{math|x}} approaches {{math|0}} or {{math|∞}}. So in the limit, the circles tend either to a limit circle or a limit point at each end. If $\backslash mu$ is a limit point or a point on the limit circle at {{math|0}} or {{math|∞}}, then $f\; =\; \backslash varphi\; +\; \backslash mu\backslash theta$ is square integrable ({{math|L²}}) near {{math|0}} or {{math|∞}}, since $\backslash mu$ lies in {{math|D_x}} for all {{math|x > c}} (in the ∞ case) and so is bounded independent of {{mvar|x}}. In particular:{{harvnb|Weyl|1910}}.{{specify|date=August 2021|reason=Which journal article, they are now differentiated.}}

• there are always non-zero solutions of {{math|1=Df = λf}} which are square integrable near {{math|0}} resp. {{math|∞}};
• in the limit circle case all solutions of {{math|1=Df = λf}} are square integrable near {{math|0}} resp. {{math|∞}}.

The radius of the disc {{math|D_x}} can be calculated to be

$$\backslash left|\{1\; \backslash over\; \{2\; \backslash operatorname\{Im\}(\backslash lambda)\; \backslash int\_c^x\; |\backslash theta|^2\}\}\backslash right|$$

and this implies that in the limit point case $\backslash theta$ cannot be square integrable near {{math|0}} resp. {{math|∞}}. 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 {{math|0}} resp. {{math|∞}}.

On the other hand, if {{math|1=Dg = λ′ g}} for another value {{math|λ′}}, then

$$h(x)\; =\; g(x)\; -(\backslash lambda^\backslash prime-\backslash lambda)\; \backslash int\_c^x\; (\backslash varphi\_\backslash lambda(x)\; \backslash theta\_\backslash lambda(y)\; -\; \backslash theta\_\backslash lambda(x)\backslash varphi\_\backslash lambda(y))g(y)\backslash ,\; dy$$

satisfies {{math|1=Dh = λh}}, so that

$$g(x\; )=\; c\_1\; \backslash varphi\_\backslash lambda\; +\; c\_2\; \backslash theta\_\backslash lambda\; +\; (\backslash lambda^\backslash prime-\backslash lambda)\; \backslash int\_c^x\; (\backslash varphi\_\backslash lambda(x)\; \backslash theta\_\backslash lambda(y)\; -\; \backslash theta\_\backslash lambda(x)\backslash varphi\_\backslash lambda(y))g(y)\backslash ,\; dy.$$

This formula may also be obtained directly by the variation of constant method from {{math|1=(D − λ)g = (λ′ − λ)g}}.

Using this to estimate {{math|g}}, it follows that

• the limit point/limit circle behaviour at {{math|0}} or {{math|∞}} is independent of the choice of {{math|λ}}.

More generally if {{math|1=Dg = (λ – r) g}} for some function {{math|r(x)}}, then{{harvnb|Bellman|1969|p=116}}

$$g(x)=c\_1\; \backslash varphi\_\backslash lambda\; +\; c\_2\; \backslash theta\_\backslash lambda\; -\; \backslash int\_c^x\; (\backslash varphi\_\backslash lambda(x)\; \backslash theta\_\backslash lambda(y)\; -\; \backslash theta\_\backslash lambda(x)\; \backslash varphi\_\backslash lambda(y))\; r(y)\; g(y)\backslash ,\; dy.$$

From this it follows that

• if {{math|r}} is continuous at {{math|0}}, then {{math|D + r}} is limit point or limit circle at {{math|0}} precisely when {{math|D}} is,

so that in particular{{harvnb|Reed|Simon|1975|p=159}}

• if {{math|q(x) − a/x²}} is continuous at {{math|0}}, then {{math|D}} is limit point at {{math|0}} if and only if {{math|a ≥ {{sfrac|3|4}}}}.

Similarly

• if {{math|r}} has a finite limit at {{math|∞}}, then {{math|D + r}} is limit point or limit circle at {{math|∞}} precisely when {{math|D}} is,

so that in particular{{harvnb|Reed|Simon|1975|p=154}}

• if {{math|q}} has a finite limit at {{math|∞}}, then {{math|D}} is limit point at {{math|∞}}.

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

Consider the differential operator

$$D\_0\; f\; =\; -(p\_0\; f\text{'})\text{'}\; +\; q\_0\; f$$

on {{open-open|0, ∞}} with {{math|q₀}} positive and continuous on {{open-open|0, ∞}} and {{math|p₀}} continuously differentiable in {{closed-open|0, ∞}}, positive in {{open-open|0, ∞}} and {{math|1=p₀(0) = 0}}.

Moreover, assume that after reduction to standard form {{math|D₀}} becomes the equivalent operator

$$Df=\; -f\text{'}\text{'}\; +\; qf$$

on {{open-open|0, ∞}} where {{math|q}} has a finite limit at {{math|∞}}. Thus

• {{math|D}} is limit point at {{math|∞}}.

At 0, {{math|D}} may be either limit circle or limit point. In either case there is an eigenfunction {{math|Φ₀}} with {{math|1=DΦ₀ = 0}} and {{math|Φ₀}} square integrable near {{math|0}}. In the limit circle case, {{math|Φ₀}} determines a boundary condition at {{math|0}}:

$$W(f,\backslash Phi\_0)(0)\; =\; 0.$$

For complex {{mvar|λ}}, let {{math|Φ_λ}} and {{math|Χ_λ}} satisfy

• {{math|1=(D – λ)Φ_λ = 0}}, {{math|1=(D – λ)Χ_λ = 0}}
• {{math|Χ_λ}} square integrable near infinity
• {{math|Φ_λ}} square integrable at {{math|0}} if {{math|0}} is limit point
• {{math|Φ_λ}} satisfies the boundary condition above if {{math|0}} is limit circle.

Let

$$\backslash omega(\backslash lambda)\; =\; W(\backslash Phi\_\backslash lambda,\backslash Chi\_\backslash lambda),$$

a constant which vanishes precisely when {{math|Φ_λ}} and {{math|Χ_λ}} are proportional, i.e. {{mvar|λ}} is an eigenvalue of {{math|D}} for these boundary conditions.

On the other hand, this cannot occur if {{math|Im λ ≠ 0}} or if {{mvar|λ}} is negative.

Indeed, if {{math|1=D f = λf}} with {{math|q₀ – λ ≥ δ > 0}}, then by Green's formula {{math|1=(Df,f) = (f,Df)}}, since {{math|W(f,f^*)}} is constant. So {{mvar|λ}} must be real. If {{math|f}} is taken to be real-valued in the {{math|D₀}} realization, then for {{math|0 < x < y}}

$$[p\_0\; f\; f\text{'}]\_x^y\; =\; \backslash int\_x^y\; (q\_0\; -\backslash lambda)|f|^2\; +\; p\_0\; (f\text{'})^2\; .$$

Since {{math|1=p₀(0) = 0}} and {{math|f}} is integrable near {{math|0}}, {{math|p₀f f′}} must vanish at {{math|0}}. Setting {{math|1=x = 0}}, it follows that {{math|f(y) f′(y) > 0}}, so that {{math|f²}} is increasing, contradicting the square integrability of {{math|f}} near {{math|∞}}.

Thus, adding a positive scalar to {{math|q}}, it may be assumed that

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

If {{math|ω(λ) ≠ 0}}, the Green's function {{math|G_λ(x,y)}} at {{mvar|λ}} is defined by

$$G\_\backslash lambda(x,y)\; =\; \backslash begin\{cases\}$$

\Phi_\lambda(x) \Chi_\lambda(y)/\omega(\lambda) & (x\le y), \\[1ex]

\Chi_\lambda(x) \Phi_\lambda(y)/\omega(\lambda) & (x\ge y).

\end{cases}

and is independent of the choice of {{math|Φ_λ}} and {{math|Χ_λ}}.

In the examples there will be a third "bad" eigenfunction {{math|Ψ_λ}} defined and holomorphic for {{mvar|λ}} not in {{closed-open|1, ∞}} such that {{math|Ψ_λ}} satisfies the boundary conditions at neither {{math|0}} nor {{math|∞}}. This means that for {{mvar|λ}} not in {{closed-open|1, ∞}}

• {{math|W(Φ_λ,Ψ_λ)}} is nowhere vanishing;
• {{math|W(Χ_λ,Ψ_λ)}} is nowhere vanishing.

In this case {{math|Χ_λ}} is proportional to {{math|Φ_λ + m(λ) Ψ_λ}}, where

$$m(\backslash lambda)\; =\; -\; W(\backslash Phi\_\backslash lambda,\; \backslash Chi\_\backslash lambda)\; /\; W(\backslash Psi\_\backslash lambda,\; \backslash Chi\_\backslash lambda).$$

Let {{math|H₁}} be the space of square integrable continuous functions on {{open-open|0, ∞}} and let {{math|H₀}} be

• the space of {{math|C²}} functions {{math|f}} on {{open-open|0, ∞}} of compact support if {{math|D}} is limit point at {{math|0}}
• the space of {{math|C²}} functions {{math|f}} on {{open-open|0, ∞}} with {{math|1=W(f, Φ₀) = 0}} at {{math|0}} and with {{math|1=f = 0}} near {{math|∞}} if {{math|D}} is limit circle at {{math|0}}.

Define {{math|1=T = G₀}} by

$$(Tf)(x)\; =\backslash int\_0^\backslash infty\; G\_0(x,y)f(y)\; \backslash ,\; dy.$$

Then {{math|1=T D = I}} on {{math|H₀}}, {{math|1=D T = I}} on {{math|H₁}} and the operator {{math|D}} is bounded below on {{math|H₀}}:

$$(Df,f)\; \backslash ge\; (f,f).$$

Thus {{math|T}} is a self-adjoint bounded operator with {{math|0 ≤ T ≤ I}}.

Formally {{math|1=T = D⁻¹}}. The corresponding operators {{math|G_λ}} defined for {{mvar|λ}} not in {{closed-open|1, ∞}} can be formally identified with

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

and satisfy {{math|1=G_λ (D – λ) = I}} on {{math|H₀}}, {{math|1=(D – λ)G_λ = I}} on {{math|H₁}}.

{{math theorem | name = Theorem.{{harvnb|Titchmarsh|1946|at=Chapter III}}{{harvnb|Kodaira|1949|pp=935–936}}

| math_statement = For every real number {{mvar|λ}} let {{math|ρ(λ)}} be defined by the Titchmarsh–Kodaira formula:

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

Then {{math|ρ(λ)}} is a lower semicontinuous non-decreasing function of {{mvar|λ}} and if

$$(Uf)(\backslash lambda)\; =\; \backslash int\_0^\backslash infty\; f(x)\; \backslash Phi(x,\backslash lambda)\; \backslash ,\; dx,$$

then {{math|U}} defines a unitary transformation of {{math|L²(0, ∞)}} onto {{math|L²({{closed-open|1,∞}}, dρ)}} such that {{math|UDU⁻¹}} corresponds to multiplication by {{mvar|λ}}.

The inverse transformation {{math|U⁻¹}} is given by

$$(U^\{-1\}g)(x)\; =\; \backslash int\_1^\backslash infty\; g(\backslash lambda)\; \backslash Phi(x,\backslash lambda)\; \backslash ,\; d\backslash rho(\backslash lambda).$$

The spectrum of {{math|D}} equals the support of {{math|dρ}}.}}

Kodaira gave a streamlined version{{harvnb|Kodaira|1949|pp=929–932}}; for omitted details, see {{harvnb|Kodaira|1950|pp=529–536}}{{harvnb|Dieudonné|1988}} of Weyl's original proof. (M.H. Stone had previously shown{{harvnb|Stone|1932|at=Chapter X}} how part of Weyl's work could be simplified using von Neumann's spectral theorem.)

In fact for {{math|1=T =D⁻¹}} with {{math|0 ≤ T ≤ I}}, the spectral projection {{math|E(λ)}} of {{math|T}} is defined by

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

It is also the spectral projection of {{math|D}} corresponding to the interval {{closed-closed|1, λ}}.

For {{math|f}} in {{math|H₁}} define

$$f(x,\backslash lambda)\; =\; (E(\backslash lambda)f)(x).$$

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

$$x\; \backslash mapsto\; (d\_\backslash lambda\; f)(x)$$

into the Banach space {{math|E}} of bounded linear functionals {{math|dρ}} on {{closed-closed|C[α, β]}} for any compact subinterval {{closed-closed|α, β}} of {{closed-open|1, ∞}}.

The functionals (or measures) {{math|d_λ f(x)}} satisfies the following {{math|E}}-valued second order ordinary differential equation:

$$D\; (d\_\backslash lambda\; f)\; =\; \backslash lambda\; \backslash cdot\; d\_\backslash lambda\; f,$$

with initial conditions at {{math|c}} in {{open-open|0, ∞}}

$$(d\_\backslash lambda\; f)(c)=\; d\_\backslash lambda\; f(c,\backslash cdot)=\backslash mu^\{(0)\},\; \backslash quad\; (d\_\backslash lambda\; f)^\backslash prime(c)=\; d\_\backslash lambda\; f\_x(c,\backslash cdot)=\backslash mu^\{(1)\}.$$

If {{math|φ_λ}} and {{math|χ_λ}} are the special eigenfunctions adapted to {{math|c}}, then

$$d\_\backslash lambda\; f\; (x)\; =\; \backslash varphi\_\backslash lambda(x)\; \backslash mu^\{(0)\}\; +\; \backslash chi\_\backslash lambda(x)\; \backslash mu^\{(1)\}.$$

Moreover,

$$\backslash mu^\{(k)\}=\; d\_\backslash lambda\; (f,\backslash xi^\{(k)\}\_\backslash lambda),$$

where

$$\backslash xi^\{(k)\}\_\backslash lambda\; =\; D\; E(\backslash lambda)\; \backslash eta^\{(k)\},$$

with

$$\backslash eta\_z^\{(0)\}(y)\; =\; G\_z(c,y),\; \backslash ,\backslash ,\backslash ,\backslash ,\; \backslash eta\_z^\{(1)\}(x)=\backslash partial\_x\; G\_z(c,y),\; \backslash ,\backslash ,\backslash ,\backslash ,\; (z\; \backslash notin\; [1,\backslash infty)).$$

(As the notation suggests, {{math|ξ_λ⁽⁰⁾}} and {{math|ξ_λ⁽¹⁾}} do not depend on the choice of {{mvar|z}}.)

Setting

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

it follows that

$$d\_\backslash lambda\; (E(\backslash lambda)\backslash eta\_z^\{(i)\},\backslash eta\_z^\{(j)\})\; =\; |\backslash lambda\; -\; z|^\{-2\}\; \backslash cdot\; d\_\backslash lambda\; \backslash sigma\_\{ij\}(\backslash lambda).$$

On the other hand, there are holomorphic functions {{math|a(λ)}}, {{math|b(λ)}} such that

• {{math|φ_λ + a(λ) χ_λ}} is proportional to {{math|Φ_λ}};
• {{math|φ_λ + b(λ) χ_λ}} is proportional to {{math|Χ_λ}}.

Since {{math|1=W(φ_λ, χ_λ) = 1}}, the Green's function is given by

$$G\_\backslash lambda(x,y)\; =\; \backslash begin\{cases\}$$

\dfrac{(\varphi_\lambda(x) + a(\lambda)\chi_\lambda(x))(\varphi_\lambda(y) + b(\lambda)\chi_\lambda(y))}{b(\lambda)-a(\lambda)} & (x\le y), \\[1ex]

\dfrac{(\varphi_\lambda(x) + b(\lambda)\chi_\lambda(x))(\varphi_\lambda(y) + a(\lambda)\chi_\lambda(y))}{b(\lambda)-a(\lambda)} & (y\le x).

\end{cases}

Direct calculation{{harvnb|Kodaira|1950|pp=534–535}} shows that

$$(\backslash eta\_z^\{(i)\},\backslash eta\_z^\{(j)\})\; =\; \backslash operatorname\{Im\}\; M\_\{ij\}(z)/\; \backslash operatorname\{Im\}\; z,$$

where the so-called {{em|characteristic matrix}} {{math|M_ij(z)}} is given by

$$M\_\{00\}(z)=\; \backslash frac\{a(z)\; b(z)\}\{a(z)\; -\; b(z)\},\backslash ,\backslash ,\; M\_\{01\}(z)\; =\; M\_\{10\}(z)\; =\; \backslash frac\{a(z)\; +\; b(z)\}\{2\; (a(z)\; -\; b(z))\},\; \backslash ,\backslash ,\; M\_\{11\}(z)\; =\; \backslash frac\{1\}\{a(z)\; -\; b(z)\}.$$

Hence

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

which immediately implies

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

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

Setting {{math|1=ψ_λ⁽⁰⁾ = φ_λ}} and {{math|1=ψ_λ⁽¹⁾ = χ_λ}}, it follows that

$$(E(\backslash mu)f)(x)\; =\; \backslash sum\_\{i.j\}\backslash int\_0^\backslash mu\; \backslash int\_0^\backslash infty\backslash psi^\{(i)\}\_\backslash lambda(x)\backslash psi^\{(j)\}\_\backslash lambda(y)\; f(y)\; \backslash ,\; dy\; \backslash ,\; d\backslash sigma\_\{ij\}(\backslash lambda)\; =\; \backslash int\_0^\backslash mu\; \backslash int\_0^\backslash infty\backslash Phi\_\backslash lambda(x)\; \backslash Phi\_\backslash lambda(y)\; f(y)\backslash ,\; dy\; \backslash ,\; d\backslash rho(\backslash lambda).$$

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

{{See also|Legendre equation|Hypergeometric equation}}

The Mehler–Fock transform{{sfn|Mehler|1881}}{{harvnb|Fock|1943|pp=253–256}}{{harvnb|Vilenkin|1968}} concerns the eigenfunction expansion associated with the Legendre differential operator {{math|D}}

$$Df\; =\; -((x^2-1)\; f\text{'})\text{'}\; =-(x^2-1)f\text{'}\text{'}\; -2x\; f\text{'}$$

on {{open-open|1, ∞}}. The eigenfunctions are the Legendre functions{{harvnb|Terras|1984|pp=261–276}}

$$P\_\{-1/2+i\backslash sqrt\{\backslash lambda\}\}(\backslash cosh\; r)\; =\; \{1\backslash over\; 2\backslash pi\}\; \backslash int\_0^\{2\backslash pi\}\; \backslash left(\{\backslash sin\; \backslash theta\; +\; i\; e^\{-r\}\; \backslash cos\backslash theta\backslash over\; \backslash cos\; \backslash theta\; -\; i\; e^\{-r\}\backslash sin\backslash theta\}\; \backslash right)^\{\{1\backslash over\; 2\}+i\backslash sqrt\{\backslash lambda\}\}\backslash ,\; d\backslash theta$$

with eigenvalue {{math|λ ≥ 0}}. The two Mehler–Fock transformations are{{harvnb|Lebedev|1972}}

$$Uf(\backslash lambda)\; =\; \backslash int\_1^\backslash infty\; f(x)\backslash ,\; P\_\{-1/2\; +i\backslash sqrt\{\backslash lambda\}\}(x)\; \backslash ,\; dx$$

and

$$U^\{-1\}g(x)=\backslash int\_0^\backslash infty\; g(\backslash lambda)\; \backslash ,\; \{1\backslash over2\}\backslash tanh\; \backslash pi\; \backslash sqrt\{\backslash lambda\}\backslash ,d\backslash lambda.$$

(Often this is written in terms of the variable {{math|1=τ = {{radic|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,{{harvnb|Vilenkin|1968|at=Chapter VI}} consider the group {{math|1=G = SU(1,1)}} consisting of complex matrices of the form

$$\backslash begin\{bmatrix\}$$

\alpha & \beta \\

\overline{\beta} & \overline{\alpha}

\end{bmatrix}

with determinant {{math|1={{abs|α}}² − {{abs|β}}² = 1}}.

{{See also|Hydrogen atom}}

A Weyl function can be defined at a singular endpoint {{math|a}} giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory.{{harvnb|Kostenko|Sakhnovich|Teschl|2012|pp=1699–1747}} this applies for example to the case of radial Schrödinger operators

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

The whole theory can also be extended to the case where the coefficients are allowed to be measures.{{harvnb|Eckhardt|Teschl|2013|pp=151–224}}

{{See also|Inverse scattering theory}}

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{{refend}}

{{Functional Analysis}}

{{SpectralTheory}}

Category:Ordinary differential equations

Category:Operator theory

Category:Spectral theory