View source for Spectral theory of ordinary differential equations

{{Short description|Part of spectral theory}}
In [[mathematics]], the '''spectral theory of ordinary differential equations''' is the part of [[spectral theory]] concerned with the determination of the [[Spectrum of an operator#Spectrum of an unbounded operator|spectrum]] and [[eigenfunction|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 operator]]s 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 theory|spectral measure]], given by the [[E. C. Titchmarsh|Titchmarsh]]–[[Kunihiko Kodaira|Kodaira]] [[Titchmarsh-Kodaira formula|formula]]. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using [[John von Neumann|von Neumann]]'s [[spectral theorem]]. It has had important applications in [[quantum mechanics]], [[operator theory]] and [[harmonic analysis]] on [[semisimple Lie group]]s.

==Introduction==
[[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 operator]]s due to [[David Hilbert]]. In his dissertation, published in 1910, [[Hermann Weyl]] extended this theory to second order ordinary differential equations with [[singularity (mathematics)|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 condition]]s 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 operator|unbounded]] [[self-adjoint operator]]s, 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 theorem|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 [[spectral theorem|resolvent]] of the singular differential operator could be approximated by [[compact operator|compact]] resolvents corresponding to [[Sturm–Liouville problem]]s 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 function|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 [[I. M. Gelfand|Gelfand]]–[[M. A. Naimark|Naimark]] can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of [[Zonal spherical function|spherical function]]s for the [[isometry group]]s of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real [[semisimple Lie group]]s 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]].

== Solutions of ordinary differential equations ==
{{Main|Ordinary differential equations}}

=== Reduction to standard form ===
Let {{math|''D''}} be the second order differential operator on {{open-open|''a'', ''b''}} given by
<math display="block"> Df(x) = -p(x) f''(x) + r(x) f'(x) + q(x) f(x),</math>
where {{math|''p''}} is a strictly positive continuously differentiable function and {{math|''q''}} and {{math|''r''}} are continuous real-valued functions.

For {{math|''x''<sub>0</sub>}} in {{open-open|''a'', ''b''}}, define the [[Sturm–Liouville theory|Liouville transformation]] {{math|''ψ''}} by
<math display="block"> \psi(x) = \int_{x_0}^x p(t)^{-1/2}\, dt</math>

If
<math display="block">U: L^2(a, b) \mapsto L^2(\psi(a), \psi(b))</math>
is the [[unitary operator]] defined by
<math display="block">(Uf)(\psi(x)) = f(x) \times \left(\psi'(x)\right)^{-1/2},\ \ \forall x \in (a, b) </math>
then
<math display="block"> U \frac{\mathrm{d}}{\mathrm{d}x} U^{-1} g = g' \psi' + \frac 1 2 g \frac{\psi''}{\psi'} </math>
and
<math display="block">\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} </math>

Hence,
<math display="block"> UDU^{-1} g= -g'' + R g' + Q g,</math>
where
<math display="block"> R = \frac { p' + r}{p^{1/2}} </math>
and
<math display="block"> Q = q - \frac {r p'}{4p} + \frac {p''}4 - \frac {5 p'^2}{16 p} </math>

The term in {{math|''g&prime;''}} can be removed using an [[Euler]] [[integrating factor]]. If {{math|1=''S&prime;''/''S'' = ''R''/2}}, then {{math|1=''h'' = ''Sg''}} satisfies
<math display="block"> (S U D U^{-1} S^{-1}) h = -h'' + V h,</math>
where the [[potential]] ''V'' is given by
<math display="block"> V = Q + \frac{S''}{S} </math>

The differential operator can thus always be reduced to one of the form <ref>{{harvnb|Titchmarsh|1962|p= 22}}</ref>
<math display="block">Df = - f'' + qf.</math>

===Existence theorem===
The following is a version of the classical [[Picard–Lindelöf theorem|Picard existence theorem]] for second order differential equations with values in a [[Banach space]] {{math|'''E'''}}.<ref>{{harvnb|Dieudonné |1969 | at= Chapter X}}</ref>

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
<math display="block">D f = A f</math>
has a unique solution {{math|''f''}} in {{math|''C''<sup>2</sup>([''a'',''b''], '''E''')}} satisfying the initial conditions
<math display="block">f(c) = \beta \, , \; f'(c) = \alpha.</math>

In fact a solution of the differential equation with these initial conditions is equivalent to a solution
of the [[integral equation]]
<math display="block">f = h + T f</math>
with {{math|''T''}} the bounded linear map on {{math|''C''([''a'',''b''], '''E''')}} defined by
<math display="block"> Tf(x) = \int_c^x K(x,y)f(y) \, dy,</math>
where {{math|''K''}} is the [[Volterra kernel]]
<math display="block">K(x, t) = (x - t) ( q(t) - A)</math>
and
<math display="block">h(x) = \alpha (x - c) + \beta.</math>

Since {{math|{{norm|''T''<sup>k</sup>}}}} tends to 0, this integral equation has a unique solution given by the [[Neumann series]]
<math display="block">f = (I - T)^{-1} h = h + T h + T^2 h + T^3 h + \cdots</math>

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

===Fundamental eigenfunctions===
If {{math|''f''}} is twice continuously differentiable (i.e. {{math|''C''<sup>2</sup>}}) 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|''C''<sup>2</sup>}} eigenfunction {{math|''f''<sub>''λ''</sub>}} on {{closed-closed|''a'', ''b''}} with {{math|''f''<sub>''λ''</sub>(''c'')}} and {{math|''f''&prime;<sub>''λ''</sub>(''c'')}} prescribed. Moreover, for each {{mvar|x}} in {{closed-closed|''a'', ''b''}}, {{math|''f''<sub>''λ''</sub>(''x'')}} and {{math|''f''&prime;<sub>''λ''</sub>(''x'')}} are [[holomorphic function]]s 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|''C''<sup>2</sup>}} eigenfunction {{math|''f''<sub>''λ''</sub>}} on {{open-open|''a'', ''b''}} with {{math|''f''<sub>''λ''</sub>(''c'')}} and {{math|''f''&prime;<sub>''λ''</sub>(''c'')}} prescribed. Moreover, for each {{mvar|x}} in {{open-open|''a'', ''b''}}, {{math|''f''<sub>''λ''</sub>(''x'')}} and {{math|''f''&prime;<sub>''λ''</sub>(''x'')}} are [[holomorphic function]]s of {{mvar|λ}}.

===Green's formula===
If {{math|''f''}} and {{math|''g''}} are {{math|''C''<sup>2</sup>}} functions on {{open-open|''a'', ''b''}}, the [[Wronskian]] {{math|''W''(''f'', ''g'')}} is defined by
<math display="block">W(f, g) (x) = f(x) g'(x) - f'(x) g(x).</math>

[[Green's identities|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''}}
<math display="block"> \int_x^y (Df) g - f (Dg) \, dt = W(f,g)(y) - W(f,g)(x).</math>

When {{math|''q''}} is continuous and {{math|''f''}}, {{math|''g''}} are {{math|''C''<sup>2</sup>}} 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
<math display="block"> \frac{d}{dx} W(f,g) = 0, </math>
so that {{math|''W''(''f'', ''g'')}} is independent of {{mvar|x}}.

== Classical Sturm–Liouville theory ==
{{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|''H''<sub>0</sub>}} be the space of {{math|''C''<sup>2</sup>}} functions {{math|''f''}} on {{closed-closed|''a'', ''b''}} satisfying the [[Robin boundary condition]]s
<math display="block">\begin{cases}
 \cos \alpha \,f(a) - \sin \alpha \,f'(a)=0, \\[0.5ex]
 \cos \beta  \,f(b) - \sin \beta \, f'(b)=0,
\end{cases}</math>
with [[inner product]]
<math display="block"> (f,g) = \int_a^b f(x) \overline{g(x)} \, dx.</math>

In practice usually one of the two standard boundary conditions:

*[[Dirichlet boundary condition]] {{math|1=''f''(''c'') = 0}}
*[[Neumann boundary condition]] {{math|1=''f''&prime;(''c'') = 0}}

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

The differential operator {{math|''D''}} given by
<math display="block"> Df = -f'' + qf </math>
acts on {{math|''H''<sub>0</sub>}}. A function {{math|''f''}} in {{math|''H''<sub>0</sub>}} 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|''H''<sub>0</sub>}}, since the Wronskian {{math|''W''(''f'', ''g'')}} vanishes if both {{math|''f''}}, {{math|''g''}} satisfy the boundary conditions:
<math display="block">(Df, g) = (f, Dg), \quad \text{ for } f, g \in H_0.</math>

As a consequence, exactly as for a [[hermitian matrix|self-adjoint matrix]] in finite dimensions,
*the eigenvalues of {{math|''D''}} are real;
*the [[eigenspace]]s for distinct eigenvalues are [[orthogonal]].

It turns out that the eigenvalues can be described by the [[Courant minimax principle|maximum-minimum]] principle of [[John Strutt, 3rd Baron Rayleigh|Rayleigh]]–[[Walter Ritz|Ritz]]<ref>{{harvnb|Courant | Hilbert |1989}}</ref> (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|''H''<sub>0</sub>}}:
{{block indent | text = <math>(Df, f) \ge M (f, f)</math> for some finite (possibly negative) constant <math>M</math>.}}

In fact, integrating by parts,
<math display="block"> (Df,f) = \left[-f' \overline{f}\right]_a^b + \int |f'|^2 + \int q |f|^2.</math>

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 inequality|Sobolev's inequality]]:

<blockquote>"Given {{math|''ε'' > 0}}, there is constant {{math|''R'' > 0}} such that {{math|{{abs|''f''(''x'')}}<sup>2</sup> ≤ ''ε'' (''f''&prime;, f&prime;) + ''R'' (''f'', ''f'')}} for all {{math|''f''}} in {{math|''C''<sup>1</sup>[''a'', ''b'']}}."</blockquote>

In fact, since
<math display="block">|f(b) - f(x)| \leq (b-a)^{1/2} \cdot \| f' \|_2,</math>
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'')<sup>''n''</sup>·(''b'' − ''a'')<sup>−''n''</sup>·''f''(''x'')}} for {{mvar|n}} sufficiently large.

=== Green's function (regular case) ===
From the theory of ordinary differential equations, there are unique fundamental eigenfunctions {{math|''φ''<sub>''λ''</sub>(''x'')}}, {{math|''χ''<sub>''λ''</sub>(''x'')}} such that

* {{math|1=''D'' ''φ''<sub>''λ''</sub> = ''λ'' ''φ''<sub>''λ''</sub>}}, {{math|1=''φ''<sub>''λ''</sub>(''a'') = sin ''α''}}, {{math|1=''φ''<sub>''λ''</sub>'(''a'') = cos ''α''}}
* {{math|1=''D'' ''χ''<sub>''λ''</sub> = ''λ'' ''χ''<sub>''λ''</sub>}}, {{math|1=''χ''<sub>''λ''</sub>(''b'') = sin ''β''}}, {{math|1=''χ''<sub>''λ''</sub>'(''b'') = cos ''β''}}

which at each point, together with their first derivatives, depend holomorphically on {{mvar|λ}}. Let
<math display="block">\omega(\lambda) = W(\phi_\lambda, \chi_\lambda),</math>
be an [[entire function|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|''H''<sub>0</sub>}}, define the [[Green's function]] by
<math display="block">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}</math>

This kernel defines an operator on the inner product space {{math|''C''[''a'',''b'']}} via
<math display="block"> (G_\lambda f)(x) =\int_a^b G_\lambda(x,y) f(y)\, dy.</math>

Since {{math|''G''<sub>''λ''</sub>(''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''] = ''H''<sub>1</sub>}} (or equivalently of the dense subspace {{math|''H''<sub>0</sub>}}), taking values in {{math|''H''<sub>1</sub>}}. This operator carries {{math|''H''<sub>1</sub>}} into {{math|''H''<sub>0</sub>}}. When {{mvar|λ}} is real, {{math|1=''G''<sub>''λ''</sub>(''x'',''y'') = ''G''<sub>''λ''</sub>(''y'',''x'')}} is also real, so defines a self-adjoint operator on {{math|''H''}}. Moreover,

* {{math|1=''G''<sub>''λ''</sub> (''D'' − ''λ'') = ''I''}} on {{math|''H''<sub>0</sub>}}
* {{math|''G''<sub>''λ''</sub>}} carries {{math|''H''<sub>1</sub>}} into {{math|''H''<sub>0</sub>}}, and {{math|1=(''D'' − ''λ'') ''G''<sub>''λ''</sub> = ''I''}} on {{math|''H''<sub>1</sub>}}.

Thus the operator {{math|''G''<sub>''λ''</sub>}} can be identified with the [[Resolvent formalism|resolvent]] {{math|(''D'' − ''λ'')<sup>−1</sup>}}.

=== Spectral theorem ===

{{math theorem | math_statement = The eigenvalues of D are real of multiplicity one and form an increasing sequence {{math|''λ''<sub>1</sub> < ''λ''<sub>2</sub> < ⋯}} tending to infinity.

The corresponding normalised eigenfunctions form an orthonormal basis of {{math|''H''<sub>0</sub>}}.

The {{mvar|k}}-th eigenvalue of {{math|''D''}} is given by the [[Courant minimax principle|minimax principle]]
<math display="block"> \lambda_k = \max_{ \dim G = k-1} \, \min_{f\perp G} {(Df,f)\over (f,f)}. </math>

In particular if {{math|''q''<sub>1</sub> ≤ ''q''<sub>2</sub>}}, then
<math display="block"> \lambda_k(D_1) \le \lambda_k(D_2).</math>}}

In fact let {{math|1=''T'' = ''G''<sub>''λ''</sub>}} for {{mvar|λ}} large and negative. Then {{math|''T''}} defines a [[compact operator|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''<sub>''n''</sub>}} of {{math|''T''}} with {{math|1=''Tψ''<sub>''n''</sub> = ''μ''<sub>''n''</sub> ''ψ''<sub>''n''</sub>}}, where {{math|''μ''<sub>''n''</sub>}} tends to zero. The range of {{math|''T''}} contains {{math|''H''<sub>0</sub>}} so is dense. Hence 0 is not an eigenvalue of {{math|''T''}}. The resolvent properties of {{math|''T''}} imply that {{math|''ψ''<sub>''n''</sub>}} lies in {{math|''H''<sub>0</sub>}} and that
<math display="block">D \psi_n = \left(\lambda + \frac{1}{\mu_n}\right) \psi_n</math>

The minimax principle follows because if
<math display="block">\lambda(G) = \min_{f\perp G} \frac{(Df,f)}{(f,f)},</math>
then {{math|1=''λ''(''G'') = ''λ''<sub>''k''</sub>}} 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''<sub>''k''</sub>}}.

=== Wronskian as a Fredholm determinant ===
For simplicity, suppose that {{math|''m'' ≤ ''q''(''x'') ≤ ''M''}} on {{closed-closed|0, ''π''}} with Dirichlet boundary conditions. The minimax principle shows that
<math display="block"> n^2 + m \le \lambda_n(D) \le n^2 + M. </math>

It follows that the resolvent {{math|(''D'' − ''λ'')<sup>−1</sup>}} 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'')<sup>−1</sup>}} is defined.

The Dirichlet boundary conditions imply that
<math display="block">\omega(\lambda) = \phi_\lambda(b).</math>

Using Picard iteration, Titchmarsh showed that {{math|''φ''<sub>''λ''</sub>(''b'')}}, and hence {{math|''ω''(''λ'')}}, is an [[entire function|entire function of finite order]] {{math|1/2}}:
<math display="block">\omega(\lambda) = \mathcal O\left(e^{\sqrt{|\lambda|}}\right)</math>

At a zero {{math|''μ''}} of {{math|''ω''(''λ'')}}, {{math|1=''φ''<sub>''μ''</sub>(''b'') = 0}}. Moreover,
<math display="block">\psi(x)=\partial_\lambda \varphi_\lambda(x)|_{\lambda=\mu}</math>
satisfies {{math|1=(''D'' − ''μ'')''ψ'' = ''φ''<sub>''μ''</sub>}}. Thus
<math display="block">\omega(\lambda) = (\lambda - \mu) \psi(b) + \mathcal O( (\lambda - \mu)^2)</math>

This implies that<ref>{{harvnb|Titchmarsh|1962}}</ref>
{{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|''H''<sub>0</sub>}}. But then
<math display="block">(\phi_\mu, \phi_\mu) = ((D - \mu)\psi, \phi_\mu) = (\psi, (D - \mu) \phi_\mu) = 0,</math>
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<ref>{{harvnb|Titchmarsh|1939|at=§8.2}}</ref>
<math display="block"> \omega(\lambda) = C \prod (1 -\lambda/\lambda_n),</math>
for some non-zero constant {{math|''C''}}.

Hence
<math display="block"> \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)} .</math>

In particular if 0 is not an eigenvalue of {{math|''D''}}
<math display="block">\omega(\mu) = \omega(0) \cdot \det ( I - \mu D^{-1}) . </math>

== Tools from abstract spectral theory ==

=== Functions of bounded variation ===
{{See also|Bounded variation|Lebesgue–Stieltjes integration|Riesz representation theorem}}
A function {{math|''ρ''(''x'')}} of [[bounded variation]]<ref>{{harvnb|Burkill|1951|pp=50–52}}</ref> 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
<math display="block"> \sum_{r=0}^{k-1} |\rho(x_{r+1}) - \rho(x_r)|</math>
over all [[partition of an interval|dissections]]
<math display="block"> a= x_0 < x_1 < \dots < x_k =b </math>
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:
<math display="block"> \rho(x) = \rho_+(x) - \rho_-(x),</math>
where {{math|''ρ''<sub>+</sub>(''x'')}} and {{math|''ρ''<sub>–</sub>(''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''}}
<math display="block">\int_a^b f(x)\, d\rho(x)</math>
is defined to be the limit of approximating sums
<math display="block">\sum_{r=0}^{k-1} f(x_r)(\rho(x_{r+1})-\rho(x_r))</math>
as the [[partition of an interval|mesh]] of the dissection, given by {{math|sup {{abs|''x''<sub>''r''+1</sub> − ''x''<sub>''r''</sub>}}}}, tends to zero.

This integral satisfies
<math display="block">\left|\int_a^b f(x)\, d\rho(x)\right|\le V(\rho)\cdot \|f\|_\infty</math>

and thus defines a [[bounded linear functional]] {{math|''dρ''}} on {{math|''C''[''a'', ''b'']}} with [[norm (mathematics)|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<ref>{{harvnb|Loomis|1953|p=[https://archive.org/details/introductiontoab031610mbp page 40]}}</ref>
<math display="block"> |\mu|(f) = \sup_{0\le |g|\le f} |\mu(g)|.</math>

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
<math display="block">|\mu(f)| \leq |\mu| (|f|)</math>
for {{math|''f''}} in {{math|''C''[''a'', ''b'']}}. If {{math|''μ''}} is ''real'', i.e. is real-valued on real-valued functions, then
<math display="block"> \mu = |\mu| - (|\mu| -\mu)\equiv \mu_+-\mu_-</math>
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 function]]s {{math|''g''}} by the formula<ref>{{harvnb|Loomis|1953|pp=30–31}}</ref>
<math display="block"> \mu(g) = \lim \mu(f_n),</math>
where the non-negative continuous functions {{math|''f''<sub>''n''</sub>}} 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<ref>{{harvnb|Kolmogorov|Fomin|1975|pp=[https://archive.org/details/introductoryreal00kolm_0/page/374 374–376]}}</ref>
<math display="block">\rho(x) = \mu(\chi_{[a,x]}),</math>
where {{math|''χ''<sub>''A''</sub>}} denotes the [[Indicator function|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''<sub>+</sub> = ''dρ''<sub>+</sub>}} and {{math|1=''μ''<sub>–</sub> = ''dρ''<sub>–</sub>}}.

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

The [[support (measure theory)|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''<sub>''A''</sub>)''f'') = 0}}, so that {{math|1=''μ''(''f'') = 0}} if {{math|''f''}} vanishes on {{math|''A''}}.

===Spectral measure===
{{See also|Spectral theorem|Projection-valued measure}}
Let {{math|''H''}} be a Hilbert space and <math>T</math> a self-adjoint [[bounded operator]] on {{math|''H''}} with <math> 0 \leq T \leq I </math>, so that the [[spectrum (functional analysis)|spectrum]] <math> \sigma(T) </math> of <math> T </math> is contained in <math>[0,1]</math>. If <math>p(t)</math> is a complex polynomial, then by the [[Continuous functional calculus#Theorem|spectral mapping theorem]]
<math display="block"> \sigma (p(T)) = p (\sigma(T)) </math>
and hence
<math display="block"> \|p(T)\| \leq \|p\|_\infty </math>
where <math> \| \cdot \|_\infty </math> 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 <math>f(T)</math> can be defined <math>\forall f \in C[0,1]</math>, with
<math display="block"> \sigma (f(T)) = f(\sigma(T))</math> and <math display="block"> \| f(T) \| \leq \|f\|_\infty.</math>

If <math>0 \leq g \leq 1 </math> is a lower semicontinuous function on {{closed-closed|0, 1}}, for example the characteristic function <math> \chi_{[0,\alpha]}</math> of a subinterval of {{closed-closed|0, 1}}, then
<math>g</math> is a pointwise increasing limit of non-negative <math>f_n \in C[0,1]</math>.

If <math>\xi</math> is a vector in {{math|''H''}}, then the vectors
<math display="block"> \eta_n = f_n(T)\xi</math>
form a [[Cauchy sequence]] in {{math|''H''}}, since, for <math>n \geq m</math>,
<math display="block"> \|\eta_n-\eta_m\|^2 \le (\eta_n,\xi) - (\eta_m,\xi),</math>
and <math>( \eta_n, \xi) = (f_n(T)\xi, \xi)</math> is bounded and increasing, so has a limit.

It follows that <math>g(T)</math> can be defined by{{efn|1=This is a limit in the [[strong operator topology]].}}
<math display="block">g(T)\xi = \lim f_n(T)\xi.</math>

If <math>\xi</math> and {{math|''η''}} are vectors in {{math|''H''}}, then
<math display="block">\mu_{\xi,\eta}(f) = (f(T) \xi,\eta)</math>
defines a bounded linear form <math>\mu_{\xi, \eta}</math> on {{math|''H''}}. By the Riesz representation theorem
<math display="block">\mu_{\xi,\eta} = d\rho_{\xi,\eta}</math>
for a unique normalised function <math>\rho_{\xi, \eta}</math> of bounded variation on {{closed-closed|0, 1}}.

<math>d\rho_{\xi, \eta}</math> (or sometimes slightly incorrectly <math>\rho_{\xi, \eta}</math> itself) is called the '''spectral measure''' determined by <math>\xi</math> and {{math|''η''}}.

The operator <math>g(T)</math> is accordingly uniquely characterised by the equation
<math display="block"> (g(T)\xi,\eta) = \mu_{\xi,\eta}(g) = \int_0^1 g(\lambda) \, d\rho_{\xi,\eta}(\lambda).</math>

The [[spectral theorem|spectral projection]] <math>E(\lambda)</math> is defined by
<math display="block">E(\lambda) = \chi_{[0,\lambda]}(T),</math>
so that
<math display="block">\rho_{\xi,\eta}(\lambda) = (E(\lambda)\xi,\eta).</math>

It follows that
<math display="block"> g(T) = \int_0^1 g(\lambda) \, dE(\lambda),</math>
which is understood in the sense that for any vectors <math>\xi</math> and <math>\eta</math>,
<math display="block"> (g(T)\xi,\eta) = \int_0^1 g(\lambda)\, d(E(\lambda)\xi,\eta) = \int_0^1 g(\lambda)\, d\rho_{\xi,\eta} (\lambda). </math>

For a single vector <math>\xi, \, \mu_{\xi} = \mu_{\xi, \xi}</math> is a positive form on {{closed-closed|0, 1}} 
(in other words proportional to a [[probability measure]] on {{closed-closed|0, 1}}) and <math>\rho_{\xi} = \rho_{\xi, \xi}</math> is non-negative and non-decreasing. Polarisation shows that all the forms <math>\mu_{\xi, \eta}</math> can naturally be expressed in terms of such positive forms, since
<math display="block">\mu_{\xi, \eta} = \frac{1}{4}\left(\mu_{\xi+\eta}+i\mu_{\xi+i\eta}-\mu_{\xi-\eta}-i\mu_{\xi-i\eta}\right)</math>

If the vector <math>\xi</math> is such that the [[linear span]] of the vectors <math>(T^n\xi)</math> is dense in {{math|''H''}}, i.e. <math>\xi</math> is a ''cyclic vector'' for <math>T</math>, then the map <math>U</math> defined by
<math display="block"> U(f) = f(T)\xi, \, C[0,1] \rightarrow H</math>
satisfies
<math display="block"> (Uf_1,Uf_2)= \int_0^1 f_1(\lambda) \overline{f_2(\lambda)} \, d\rho_\xi(\lambda).</math>

Let <math>L_2([0,1], d\rho_\xi)</math> denote the Hilbert space completion of <math>C[0,1]</math> associated with the possibly [[inner product#Degenerate inner products|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 <math>f</math>, i.e. those with <math>\mu_\xi(|f|_2) = 0</math>. Alternatively in this case the support of the measure is <math>\sigma(T)</math>, so the right hand side defines a (non-degenerate) inner product on <math>C(\sigma(T))</math>.}} 
Thus <math>U</math> extends to a [[unitary transformation]] of <math>L_2([0,1], \rho_\xi)</math> onto {{math|''H''}}. <math>U T U^\ast</math> is then just multiplication by <math>\lambda</math> on <math>L_2([0,1], d\rho_\xi)</math>; and more generally <math>U f(T) U^\ast</math> is multiplication by <math>f(\lambda)</math>. In this case, the support of <math>d\rho_\xi</math> is exactly <math>\sigma(T)</math>, 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''.}}

== Weyl–Titchmarsh–Kodaira theory ==

The eigenfunction expansion associated with singular differential operators of the form
<math display="block"> Df = -(pf')' + qf</math>
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 condition]]s there. Unlike the regular Sturm–Liouville case, in some circumstances [[spectral theory|spectral values]] of {{math|''D''}} can have [[Multiplicity (mathematics)|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'')<sup>−1</sup>}} 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''<sup>−1</sup>}} and let
<math display="block"> E(\lambda) = \chi_{[\lambda^{-1},1]}(T)</math>
be the spectral projection of {{math|''D''}} corresponding to the interval {{closed-closed|1, ''λ''}}. For an arbitrary function {{math|''f''}} define
<math display="block"> f(x,\lambda) = (E(\lambda)f)(x). </math>
{{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
<math display="block"> x\mapsto (d_\lambda f)(x)</math>
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''<sub>''λ''</sub> ''f''}} satisfies a second order ordinary differential equation taking values in {{math|'''E'''}}:
<math display="block"> D (d_\lambda f) = \lambda \cdot d_\lambda f.</math>

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
<math display="block"> (d_\lambda f)(c) = d_\lambda f(c,\cdot), \quad (d_\lambda f)^\prime(c) = d_\lambda f_x(c,\cdot).</math>

This point of view may now be turned on its head: {{math|''f''(''c'', ''λ'')}} and {{math|''f''<sub>''x''</sub>(''c'', ''λ'')}} may be written as
<math display="block"> f(c,\lambda) = (f,\xi_1(\lambda)), \quad f_x(c,\lambda) = (f,\xi_2(\lambda)),</math>
where {{math|''ξ''<sub>1</sub>(''λ'')}} and {{math|''ξ''<sub>2</sub>(''λ'')}} are given purely in terms of the fundamental eigenfunctions.
The functions of bounded variation
<math display="block"> \sigma_{ij}(\lambda) = (\xi_i(\lambda),\xi_j(\lambda))</math>
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).

=== Limit circle and limit point for singular equations ===
Let {{math|''q''(''x'')}} be a continuous real-valued function on {{open-open|0, ∞}} and let {{math|''D''}} be the second order differential operator
<math display="block"> Df = -f'' + qf</math>
on {{open-open|0, ∞}}. Fix a point {{math|''c''}} in {{open-open|0, ∞}} and, for complex {{mvar|λ}}, let <math>\varphi_\lambda, \theta_\lambda</math> be the unique '''fundamental eigenfunctions''' of {{math|''D''}} on {{open-open|0, ∞}} satisfying
<math display="block"> (D-\lambda)\varphi_\lambda = 0, \quad (D-\lambda)\theta_\lambda =0 </math>
together with the initial conditions at {{math|''c''}}
<math display="block"> \varphi_\lambda(c) = 1,\, \varphi_\lambda'(c) = 0, \, \theta_\lambda(c) = 0, \, \theta_\lambda'(c) = 1.</math>

Then their Wronskian satisfies
<math display="block"> W(\varphi_\lambda,\theta_\lambda) = \varphi_\lambda\theta_\lambda'- \theta_\lambda \varphi_\lambda' \equiv 1,</math>

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 <math>\mu</math> is such that <math>f = \varphi + \mu \theta</math> satisfies the boundary condition <math>\cos\beta\, f(x) - \sin\beta\, f'(x) = 0</math> for some <math>\beta</math> (or, equivalently, <math>f'(x) / f(x)</math> is real) then, using integration by parts, one obtains
<math display="block"> \operatorname{Im}(\lambda) \int_c^x |\varphi +\mu \theta|^2 = \operatorname{Im}(\mu).</math>

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
<math display="block"> \int_c^x | \varphi + \mu \theta|^2 < {\operatorname{Im}(\mu) \over \operatorname{Im}(\lambda)}</math>
if {{math|''x'' > ''c''}} and by
<math display="block"> \int_x^c | \varphi + \mu \theta|^2 < {\operatorname{Im}(\mu) \over \operatorname{Im}(\lambda)}</math>
if {{math|''x'' < ''c''}}.

Let {{math|''D''<sub>''x''</sub>}} 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 <math>\mu</math> is a limit point or a point on the limit circle at {{math|0}} or {{math|∞}}, then <math>f = \varphi + \mu\theta</math> is [[square integrable]] ({{math|''L''<sup>2</sup>}}) near {{math|0}} or {{math|∞}}, since <math>\mu</math> lies in {{math|''D''<sub>''x''</sub>}} for all {{math|''x'' > ''c''}} (in the ∞ case) and so <math> \int_c^x | \varphi +\mu \theta|^2 < { \operatorname{Im}(\mu) \over \operatorname{Im}(\lambda)}</math> is bounded independent of {{mvar|x}}. In particular:<ref name="Weyl 1910">{{harvnb|Weyl|1910}}.{{specify|date=August 2021|reason=Which journal article, they are now differentiated.}}</ref>

* 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''<sub>''x''</sub>}} can be calculated to be 
<math display="block"> \left|{1 \over {2 \operatorname{Im}(\lambda) \int_c^x |\theta|^2}}\right|</math>
and this implies that in the limit point case <math>\theta</math> 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''&prime; ''g''}} for another value {{math|''λ''&prime;}}, then
<math display="block"> 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</math>
satisfies {{math|1=''Dh'' = ''λh''}}, so that
<math display="block">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.</math>

This formula may also be obtained directly by the variation of constant method from {{math|1=(''D'' − ''λ'')''g'' = (''λ''&prime; − ''λ'')''g''}}.
Using this to estimate {{math|''g''}}, it follows that<ref name="Weyl 1910"/>

* 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<ref name="Bellman 1969 116">{{harvnb|Bellman|1969|p=116}}</ref>
<math display="block">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.</math>

From this it follows that<ref name="Bellman 1969 116"/>

* 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<ref>{{harvnb|Reed|Simon|1975|p=159}}</ref>

* if {{math|''q''(''x'') − ''a''/''x''<sup>2</sup>}} 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<ref>{{harvnb|Reed|Simon|1975|p=154}}</ref>

* 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.

=== Green's function (singular case) ===
Consider the differential operator
<math display="block"> D_0 f = -(p_0 f')' + q_0 f</math>
on {{open-open|0, ∞}} with {{math|''q''<sub>0</sub>}} positive and continuous on {{open-open|0, ∞}} and {{math|''p''<sub>0</sub>}} continuously differentiable in {{closed-open|0, ∞}}, positive in {{open-open|0, ∞}} and {{math|1=''p''<sub>0</sub>(0) = 0}}.

Moreover, assume that after reduction to standard form {{math|''D''<sub>0</sub>}} becomes the equivalent operator
<math display="block"> Df= -f'' + qf</math>
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|Φ<sub>0</sub>}} with {{math|1=''D''Φ<sub>0</sub> = 0}} and {{math|Φ<sub>0</sub>}} square integrable near {{math|0}}. In the limit circle case, {{math|Φ<sub>0</sub>}} determines a '''[[boundary condition]]''' at {{math|0}}:
<math display="block"> W(f,\Phi_0)(0) = 0.</math>

For complex {{mvar|λ}}, let {{math|Φ<sub>''λ''</sub>}} and {{math|Χ<sub>''λ''</sub>}} satisfy

* {{math|1=(''D'' – ''λ'')Φ<sub>''λ''</sub> = 0}}, {{math|1=(''D'' – ''λ'')Χ<sub>''λ''</sub> = 0}}
* {{math|Χ<sub>''λ''</sub>}} square integrable near infinity
* {{math|Φ<sub>''λ''</sub>}} square integrable at {{math|0}} if {{math|0}} is ''limit point''
* {{math|Φ<sub>''λ''</sub>}} satisfies the boundary condition above if {{math|0}} is ''limit circle''.

Let
<math display="block">\omega(\lambda) = W(\Phi_\lambda,\Chi_\lambda),</math>
a constant which vanishes precisely when {{math|Φ<sub>''λ''</sub>}} and {{math|Χ<sub>''λ''</sub>}} 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.<ref name="Weyl 1910"/>

Indeed, if {{math|1=''D f'' = ''λf''}} with {{math|''q''<sub>0</sub> – ''λ'' ≥ ''δ'' > 0}}, then by Green's formula {{math|1=(''Df'',''f'') = (''f'',''Df'')}}, since {{math|''W''(''f'',''f''<sup>*</sup>)}} is constant. So {{mvar|λ}} must be real. If {{math|''f''}} is taken to be real-valued in the {{math|''D''<sub>0</sub>}} realization, then for {{math|0 < ''x'' < ''y''}}
<math display="block"> [p_0 f f']_x^y = \int_x^y (q_0 -\lambda)|f|^2 + p_0 (f')^2 .</math>

Since {{math|1=''p''<sub>0</sub>(0) = 0}} and {{math|''f''}} is integrable near {{math|0}}, {{math|''p''<sub>0</sub>''f'' ''f''&prime;}} must vanish at {{math|0}}. Setting {{math|1=''x'' = 0}}, it follows that {{math|''f''(''y'') ''f''&prime;(''y'') > 0}}, so that {{math|''f''<sup>2</sup>}} is increasing, contradicting the square integrability of {{math|''f''}} near {{math|∞}}.

Thus, adding a positive scalar to {{math|''q''}}, it may be assumed that
<math display="block">\omega(\lambda) \neq 0 ~~ \text{ if } \lambda \notin [1, \infty).</math>

If {{math|''ω''(''λ'') ≠ 0}}, the '''[[Green's function]]''' {{math|''G''<sub>''λ''</sub>(''x'',''y'')}} at {{mvar|λ}} is defined by
<math display="block">G_\lambda(x,y) = \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}</math>
and is independent of the choice of {{math|Φ<sub>''λ''</sub>}} and {{math|Χ<sub>''λ''</sub>}}.

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

* {{math|''W''(Φ<sub>''λ''</sub>,Ψ<sub>''λ''</sub>)}} is nowhere vanishing;
* {{math|''W''(Χ<sub>''λ''</sub>,Ψ<sub>''λ''</sub>)}} is nowhere vanishing.

In this case {{math|Χ<sub>''λ''</sub>}} is proportional to {{math|Φ<sub>''λ''</sub> + ''m''(''λ'') Ψ<sub>''λ''</sub>}}, where
<math display="block">m(\lambda) = - W(\Phi_\lambda, \Chi_\lambda) / W(\Psi_\lambda, \Chi_\lambda).</math>

Let {{math|''H''<sub>1</sub>}} be the space of square integrable continuous functions on {{open-open|0, ∞}} and let {{math|''H''<sub>0</sub>}} be

* the space of {{math|''C''<sup>2</sup>}} functions {{math|''f''}} on {{open-open|0, ∞}} of [[compact support]] if {{math|''D''}} is limit point at {{math|0}}
* the space of {{math|''C''<sup>2</sup>}} functions {{math|''f''}} on {{open-open|0, ∞}} with {{math|1=''W''(''f'', Φ<sub>0</sub>) = 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'' = ''G''<sub>0</sub>}} by
<math display="block">(Tf)(x) =\int_0^\infty G_0(x,y)f(y) \, dy. </math>

Then {{math|1=''T'' ''D'' = ''I''}} on {{math|''H''<sub>0</sub>}}, {{math|1=''D'' ''T'' = ''I''}} on {{math|''H''<sub>1</sub>}} and the operator {{math|''D''}} is bounded below on {{math|''H''<sub>0</sub>}}:
<math display="block">(Df,f) \ge (f,f).</math>

Thus {{math|''T''}} is a self-adjoint bounded operator with {{math|0 ≤ ''T'' ≤ ''I''}}.

Formally {{math|1=''T'' = ''D''<sup>−1</sup>}}. The corresponding operators {{math|''G''<sub>''λ''</sub>}} defined for {{mvar|λ}} not in {{closed-open|1, ∞}} can be formally identified with
<math display="block">(D-\lambda)^{-1} = T(I-\lambda T)^{-1}</math>
and satisfy {{math|1=''G''<sub>''λ''</sub> (''D'' – ''λ'') = ''I''}} on {{math|''H''<sub>0</sub>}}, {{math|1=(''D'' – ''λ'')''G''<sub>''λ''</sub> = ''I''}} on {{math|''H''<sub>1</sub>}}.

=== Spectral theorem and Titchmarsh–Kodaira formula ===

{{math theorem | name = Theorem.<ref name="Weyl 1910"/><ref>{{harvnb|Titchmarsh|1946|at=Chapter III}}</ref><ref>{{harvnb|Kodaira|1949|pp=935–936}}</ref>
| math_statement = For every real number {{mvar|λ}} let {{math|''ρ''(''λ'')}} be defined by the '''Titchmarsh–Kodaira formula''':
<math display="block"> \rho(\lambda) = \lim_{\delta \downarrow 0} \lim_{\varepsilon \downarrow 0} \frac{1}{\pi} \int_\delta^{\lambda+\delta} \operatorname{Im} m(t + i\varepsilon) \, dt.</math>

Then {{math|''ρ''(''λ'')}} is a lower semicontinuous non-decreasing function of {{mvar|λ}} and if
<math display="block"> (Uf)(\lambda) = \int_0^\infty f(x) \Phi(x,\lambda) \, dx,</math>
then {{math|''U''}} defines a unitary transformation of {{math|''L''<sup>2</sup>(0, ∞)}} onto {{math|''L''<sup>2</sup>({{closed-open|1,∞}}, ''dρ'')}} such that {{math|''UDU''<sup>−1</sup>}} corresponds to multiplication by {{mvar|λ}}. 

The inverse transformation {{math|''U''<sup>−1</sup>}} is given by
<math display="block"> (U^{-1}g)(x) = \int_1^\infty g(\lambda) \Phi(x,\lambda) \, d\rho(\lambda).</math>

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

Kodaira gave a streamlined version<ref>{{harvnb|Kodaira|1949|pp=929–932}}; for omitted details, see {{harvnb|Kodaira|1950|pp=529–536}}</ref><ref>{{harvnb|Dieudonné|1988}}</ref> of Weyl's original proof.<ref name="Weyl 1910"/> ([[Marshall Harvey Stone|M.H. Stone]] had previously shown<ref>{{harvnb|Stone|1932|at=Chapter X}}</ref> how part of Weyl's work could be simplified using von Neumann's spectral theorem.)

In fact for {{math|1=''T'' =''D''<sup>−1</sup>}} with {{math|0 ≤ ''T'' ≤ ''I''}}, the spectral projection {{math|''E''(''λ'')}} of {{math|''T''}} is defined by
<math display="block"> E(\lambda) =\chi_{[\lambda^{-1},1]}(T)</math>

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

For {{math|''f''}} in {{math|''H''<sub>1</sub>}} define
<math display="block"> f(x,\lambda) = (E(\lambda)f)(x). </math>

{{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
<math display="block"> x \mapsto (d_\lambda f)(x)</math>
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''<sub>''λ''</sub> ''f''(''x'')}} satisfies the following {{math|'''E'''}}-valued second order ordinary differential equation:
<math display="block"> D (d_\lambda f) = \lambda \cdot d_\lambda f,</math>
with initial conditions at {{math|''c''}} in {{open-open|0, ∞}}
<math display="block"> (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)}.</math>

If {{math|''φ''<sub>''λ''</sub>}} and {{math|''χ''<sub>''λ''</sub>}} are the special eigenfunctions adapted to {{math|''c''}}, then
<math display="block"> d_\lambda f (x) = \varphi_\lambda(x) \mu^{(0)} + \chi_\lambda(x) \mu^{(1)}.</math>

Moreover,
<math display="block"> \mu^{(k)}= d_\lambda (f,\xi^{(k)}_\lambda),</math>
where
<math display="block"> \xi^{(k)}_\lambda = D E(\lambda) \eta^{(k)},</math>
with
<math display="block"> \eta_z^{(0)}(y) = G_z(c,y), \,\,\,\, \eta_z^{(1)}(x)=\partial_x G_z(c,y), \,\,\,\, (z \notin [1,\infty)).</math>
(As the notation suggests, {{math|''ξ''<sub>''λ''</sub><sup>(0)</sup>}} and {{math|''ξ''<sub>''λ''</sub><sup>(1)</sup>}} do not depend on the choice of {{mvar|z}}.)

Setting
<math display="block">\sigma_{ij}(\lambda) = (\xi^{(i)}_\lambda, \xi^{(j)}_\lambda),</math>
it follows that
<math display="block"> d_\lambda (E(\lambda)\eta_z^{(i)},\eta_z^{(j)}) = |\lambda - z|^{-2} \cdot d_\lambda \sigma_{ij}(\lambda).</math>

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

* {{math|''φ''<sub>''λ''</sub> + ''a''(''λ'') ''χ''<sub>''λ''</sub>}} is proportional to {{math|Φ<sub>''λ''</sub>}};
* {{math|''φ''<sub>''λ''</sub> + ''b''(''λ'') ''χ''<sub>''λ''</sub>}} is proportional to {{math|Χ<sub>''λ''</sub>}}.

Since {{math|1=''W''(''φ''<sub>''λ''</sub>, ''χ''<sub>''λ''</sub>) = 1}}, the Green's function is given by
<math display="block"> 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\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} </math>

Direct calculation<ref>{{harvnb|Kodaira|1950|pp=534–535}}</ref> shows that
<math display="block">(\eta_z^{(i)},\eta_z^{(j)}) = \operatorname{Im} M_{ij}(z)/ \operatorname{Im} z, </math>
where the so-called {{em|'''characteristic matrix'''}} {{math|''M''<sub>''ij''</sub>(''z'')}} is given by
<math display="block"> 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)}.</math>

Hence
<math display="block"> \int_{-\infty}^\infty (\operatorname{Im} z)\cdot|\lambda-z|^{-2}\, d\sigma_{ij}(\lambda) = \operatorname{Im} M_{ij}(z),</math>
which immediately implies
<math display="block"> \sigma_{ij}(\lambda) = \lim_{\delta\downarrow 0} \lim_{\varepsilon \downarrow 0} \int_\delta^{\lambda +\delta} \operatorname{Im} M_{ij}(t +i\varepsilon)\, dt.</math>
(This is a special case of the [[Stieltjes transformation|"Stieltjes inversion formula"]].)

Setting {{math|1=''ψ''<sub>''λ''</sub><sup>(0)</sup> = ''φ''<sub>''λ''</sub>}} and {{math|1=''ψ''<sub>''λ''</sub><sup>(1)</sup> = ''χ''<sub>''λ''</sub>}}, it follows that
<math display="block"> (E(\mu)f)(x) = \sum_{i.j}\int_0^\mu \int_0^\infty\psi^{(i)}_\lambda(x)\psi^{(j)}_\lambda(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).</math>

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

== Application to the hypergeometric equation ==
{{See also|Legendre equation|Hypergeometric equation}}
The '''Mehler–Fock transform'''{{sfn|Mehler|1881}}<ref>{{harvnb|Fock|1943|pp=253–256}}</ref><ref>{{harvnb|Vilenkin|1968}}</ref> concerns the eigenfunction expansion associated with the [[Legendre function|Legendre differential operator]] {{math|''D''}}
<math display="block"> Df = -((x^2-1) f')' =-(x^2-1)f'' -2x f' </math>
on {{open-open|1, ∞}}. The eigenfunctions are the [[Legendre function]]s<ref>{{harvnb|Terras|1984|pp=261–276}}</ref>
<math display="block"> P_{-1/2+i\sqrt{\lambda}}(\cosh r) = {1\over 2\pi} \int_0^{2\pi} \left({\sin \theta + i e^{-r} \cos\theta\over \cos \theta - i e^{-r}\sin\theta} \right)^{{1\over 2}+i\sqrt{\lambda}}\, d\theta</math>
with eigenvalue {{math|''λ'' ≥ 0}}. The two Mehler–Fock transformations are<ref>{{harvnb|Lebedev|1972}}</ref>
<math display="block">Uf(\lambda) = \int_1^\infty f(x)\, P_{-1/2 +i\sqrt{\lambda}}(x) \, dx</math>
and
<math display="block">U^{-1}g(x)=\int_0^\infty g(\lambda) \, {1\over2}\tanh \pi \sqrt{\lambda}\,d\lambda.</math>

(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,<ref>{{harvnb|Vilenkin|1968|at=Chapter VI}}</ref> consider the group {{math|1=''G'' = [[SL(2,R)|SU(1,1)]]}} consisting of complex matrices of the form
<math display="block"> \begin{bmatrix}
\alpha & \beta \\
\overline{\beta} & \overline{\alpha}
\end{bmatrix}</math>

with determinant {{math|1={{abs|''α''}}<sup>2</sup> − {{abs|''β''}}<sup>2</sup> = 1}}.

== Application to the hydrogen atom ==
{{See also|Hydrogen atom}}

== Generalisations and alternative approaches ==
A Weyl function can be defined at a singular endpoint {{math|''a''}} giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory.<ref>{{harvnb|Kostenko|Sakhnovich|Teschl|2012|pp=1699–1747}}</ref> this applies for example to the case of radial Schrödinger operators
<math display="block"> Df = -f'' + \frac{\ell(\ell+1)}{x^2} f+ V(x) f, \qquad x\in(0,\infty) </math>

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

== Gelfand–Levitan theory ==
{{See also|Inverse scattering theory}}

==Notes==
{{notelist}}

== References ==
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{{refend}}

{{Functional Analysis}}
{{SpectralTheory}}

[[Category:Ordinary differential equations]]
[[Category:Operator theory]]
[[Category:Spectral theory]]