KNOWPIA
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- Summary
- Introduction
- Solutions of ordinary differential equations
- Classical Sturm–Liouville theory
- Tools from abstract spectral theory
- Weyl–Titchmarsh–Kodaira theory
- Application to the hypergeometric equation
- Application to the hydrogen atom
- Generalisations and alternative approaches
- Gelfand–Levitan theory
- Notes

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

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.

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

where *p* is a strictly positive continuously differentiable function and *q* and *r* are continuous
real-valued functions.

For *x*_{0} in (*a*, *b*), define the Liouville transformation ψ by

If

is the unitary operator defined by

then

and

Hence,

where

and

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

where the potential

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

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

Let α, β be arbitrary elements of **E**, *A* a bounded operator on *E* and *q* a continuous function on [*a*, *b*].

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

*Df*=*Af*

has a unique solution *f* in *C*^{2}([*a*,*b*],**E**) satisfying the initial conditions

*f*(*c*) = β,*f*'(*c*) = α.

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 *T* the bounded linear map on *C*([*a*,*b*], **E**) defined by

where *K* is the Volterra kernel

*K*(*x*,*t*)= (*x*−*t*)(*q*(*t*) −*A*)

and

*h*(*x*) = α(*x*−*c*) + β.

Since ||*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*+ ⋯

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

If *f* is twice continuously differentiable (i.e. *C*^{2}) on (*a*, *b*) satisfying *Df* = λ*f*, then *f* is called an eigenfunction of *L* with eigenvalue λ.

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

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

*W*(*f*,*g*) (x) =*f*(*x*)*g*'(*x*) −*f*'(*x*)*g*(*x*).

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

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

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

so that

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

with inner product

In practise usually one of the two standard boundary conditions:

- Dirichlet boundary condition
*f*(*c*) = 0 - Neumann boundary condition
*f*'(*c*) = 0

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

The differential operator *D* given by

acts on

- (
*Df*,*g*) = (*f*,*Dg*) for*f*,*g*in*H*_{0}.

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

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

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

- for some finite (possibly negative) constant .

In fact integrating by parts

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

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

- "
*Given ε > 0, there is constant R >0 such that |f(x)|*^{2}≤ ε*(f', f') + R (f, f) for all f in C*"^{1}[a, b].

- "

In fact, since

- |
*f*(*b*) −*f*(*x*)| ≤ (*b*−*a*)^{1/2}·||*f*'||_{2},

- |

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

From the theory of ordinary differential equations, there are unique fundamental eigenfunctions φ_{λ}(x), χ_{λ}(x) such that

*D*φ_{λ}= λ φ_{λ}, φ_{λ}(*a*) = sin α, φ_{λ}'(*a*) = cos α*D*χ_{λ}= λ χ_{λ}, χ_{λ}(*b*) = sin β, χ_{λ}'(*b*) = cos β

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

- ω(λ) = W(φ
_{λ}, χ_{λ}),

be an entire holomorphic function.

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

If λ is not an eigenvalue of *D* on *H*_{0}, define the Green's function by

*G*_{λ}(*x*,*y*) = φ_{λ}(*x*) χ_{λ}(*y*) / ω(λ) for*x*≥*y*and χ_{λ}(*x*) φ_{λ}(*y*) / ω(λ) for*y*≥*x*.

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

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

*G*_{λ}(*D*− λ) =I on*H*_{0}*G*_{λ}carries*H*_{1}into*H*_{0}, and (*D*− λ)*G*_{λ}= I on*H*_{1}.

Thus the operator *G*_{λ} can be identified with the resolvent (*D* − λ)^{−1}.

**Theorem.** *The eigenvalues of D are real of multiplicity one and form an increasing sequence λ _{1} < λ_{2} < ··· tending to infinity.*

*The corresponding normalised eigenfunctions form an orthonormal basis of* *H*_{0}.

*The kth eigenvalue of D is given by the minimax principle*

*In particular if q _{1} ≤ q_{2}, then*

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

*D*ψ_{n}= (λ + 1/μ_{n}) ψ_{n}

The minimax principle follows because if

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

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

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

The Dirichlet boundary conditions imply that

- ω(λ)= φ
_{λ}(*b*).

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

- ω(λ) = O(e
^{√|λ|})

At a zero μ of ω(λ), φ_{μ}(*b*) = 0. Moreover,

satisfies (

- ω(λ) = (λ − μ)ψ(
*b*) + O( (λ − μ)^{2}).

This implies that^{[4]}

- μ is a simple zero of ω(λ).

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

- (φ
_{μ}, φ_{μ}) = ((*D*− μ)ψ, φ_{μ}) = (ψ, (*D*− μ)φ_{μ}) = 0,

a contradiction.

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

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

for some non-zero constant *C*.

Hence

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

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

over all dissections

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

where ρ

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

is defined to be the limit of approximating sums

as the mesh of the dissection, given by sup |

This integral satisfies

and thus defines a bounded linear functional *d*ρ on *C*[*a*, *b*] with norm ||d*ρ||=*V*(ρ).*

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

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

- |μ(
*f*)| ≤ |μ|(|*f*|)

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

gives a canonical decomposition as a difference of

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

where the non-negative continuous functions

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

where χ

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

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

Let *H* be a Hilbert space and a self-adjoint bounded operator on *H* with , so that the spectrum of is contained in . If is a complex polynomial, then by the *spectral mapping theorem*

and hence

where denotes the uniform norm on . By the Weierstrass approximation theorem, polynomials are uniformly dense in . It follows that can be defined , with

and .

If is a lower semicontinuous function on , for example the characteristic function of a subinterval of , then is a pointwise increasing limit of non-negative .

According to Szőkefalvi-Nagy,^{[10]} if is a vector in *H*, then the vectors

form a Cauchy sequence in

and is bounded and increasing, so has a limit.

It follows that can be defined by^{[a]}

If and are vectors in *H*, then

defines a bounded linear form on

for a unique normalised function of bounded variation on .

(or sometimes slightly incorrectly itself) is called the **spectral measure**
determined by and .

The operator is accordingly uniquely characterised by the equation

The spectral projection is defined by

so that

It follows that

which is understood in the sense that for any vectors and ,

For a single vector is a positive form on (in other words proportional to a probability measure on ) and is non-negative and non-decreasing. Polarisation shows that all the forms can naturally be expressed in terms of such positive forms, since

If the vector is such that the linear span of the vectors is dense in *H*, i.e. is a *cyclic vector* for , then the map defined by

satisfies

Let denote the Hilbert space completion of associated
with the possibly degenerate inner product on the right hand side.^{[b]}
Thus extends to a unitary transformation of onto *H*. is then just multiplication by on ; and more generally is multiplication by . In this case, the support of
is exactly , so that

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

The eigenfunction expansion associated with singular differential operators of the form

on an open interval (

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

The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of *D* lies in [1,∞) and that *T* =*D*^{−1} and let

be the spectral projection of

into the Banach space

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

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

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

where ξ

determine a spectral measure on the spectrum of

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

on (0,∞). Fix a point

together with the initial conditions at

Then their Wronskian satisfies

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

Let λ be non-real and 0 < *x* < ∞. If the complex number is such that satisfies the boundary condition for some (or, equivalently, is real) then, using integration by parts, one obtains

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

if

if

Let *D*_{x} be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as *x* approaches 0 or ∞. So in the limit, the circles tend either to a **limit circle** or a **limit point** at each end. If is a limit point or a point on the limit circle at 0 or ∞, then is square integrable (L^{2}) near 0 or ∞, since lies in *D*_{x} for all *x>c* (in the ∞ case) and so is bounded independent of *x*. In particular:^{[11]}

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

The radius of the disc *D*_{x} can be calculated to be

and this implies that in the limit point case cannot be square integrable near 0 resp. ∞. Therefore, we have a converse to the second statement above:

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

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

satisfies

This formula may also be obtained directly by the variation of constant method from (D-λ)g = (λ'-λ)g.
Using this to estimate *g*, it follows that^{[11]}

*the limit point/limit circle behaviour at 0 or ∞ is independent of the choice of λ*.

More generally if *Dg*= (λ – *r*) *g* for some function *r*(*x*), then^{[12]}

From this it follows that^{[12]}

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

so that in particular^{[13]}

*if q(x)- a/x*.^{2}is continuous at 0, then D is limit point at 0 if and only if a ≥ ¾

Similarly

*if r has a finite limit at ∞, then D + r is limit point or limit circle at ∞ precisely when D is*,

so that in particular^{[14]}

*if q has a finite limit at ∞, then D is limit point at ∞*.

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

Consider the differential operator

on (0,∞) with *q*_{0} positive and continuous on (0,∞) and *p*_{0} continuously differentiable in [0,∞), positive in (0,∞) and *p*_{0}(0)=0.

Moreover, assume that after reduction to standard form
*D*_{0} becomes the equivalent operator

on (0,∞) where *q* has a finite limit at ∞. Thus

*D is limit point at ∞*.

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

For λ complex, let Φ_{λ} and Χ_{λ} satisfy

- (
*D*– λ)Φ_{λ}= 0, (*D*– λ)Χ_{λ}= 0 - Χ
_{λ}square integrable near infinity - Φ
_{λ}square integrable at 0 if 0 is*limit point* - Φ
_{λ}satisfies the boundary condition above if 0 is*limit circle*.

Let

a constant which vanishes precisely when Φ

On the other hand, this cannot occur if Im λ ≠ 0 or if λ is negative.^{[11]}

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

Since *p*_{0}(0) = 0 and *f* is integrable near 0, *p*_{0}*f* *f* ' must vanish at 0. Setting *x* = 0, it follows that *f*(*y*) *f* '(*y*) >0, so that *f*^{2} is increasing, contradicting the square integrability of *f* near ∞.

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

*ω(λ) ≠ 0 when λ is not in [1,∞)*.

If ω(λ) ≠ 0, the **Green's function** *G*_{λ}(*x*,*y*) at λ is defined by

and is independent of the choice of

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

*W*(Φ_{λ},Ψ_{λ}) is nowhere vanishing;*W*(Χ_{λ},Ψ_{λ}) is nowhere vanishing.

In this case Χ_{λ} is proportional to Φ_{λ} + *m*(λ) Ψ_{λ}, where

*m*(λ) = –*W*(Φ_{λ},Χ_{λ}) /*W*(Ψ_{λ},Χ_{λ}).

Let *H*_{1} be the space of square integrable continuous functions on (0,∞) and let *H*_{0} be

- the space of C
^{2}functions*f*on (0,∞) of compact support if*D*is limit point at 0 - the space of C
^{2}functions*f*on (0,∞) with*W*(*f*,Φ_{0})=0 at 0 and with*f*= 0 near ∞ if*D*is limit circle at 0.

Define *T* = *G*_{0} by

Then *T* *D* = *I* on *H*_{0}, *D* *T* = *I* on *H*_{1} and the operator *D* is bounded below on *H*_{0}:

Thus *T* is a self-adjoint bounded operator with 0 ≤ *T* ≤ *I*.

Formally *T* = *D*^{−1}. The corresponding operators *G*_{λ} defined for λ not in [1,∞) can be formally identified with

and satisfy

**Theorem**.^{[11]}^{[15]}^{[16]} *For every real number λ let ρ(λ) be defined by the* **Titchmarsh–Kodaira formula**:

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

*The inverse transformation U ^{−1} is given by*

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

Kodaira gave a streamlined version^{[17]}^{[18]} of Weyl's original proof.^{[11]} (M.H. Stone had previously shown^{[19]} how part of Weyl's work could be simplified using von Neumann's spectral theorem.)

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

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

For *f* in *H*_{1} define

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

into the Banach space

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

with initial conditions at

If φ_{λ} and χ_{λ} are the special eigenfunctions adapted to *c*, then

Moreover,

where

with

(As the notation suggests, ξ

Setting

it follows that

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

- φ
_{λ}+*a*(λ) χ_{λ}is proportional to Φ_{λ}; - φ
_{λ}+*b*(λ) χ_{λ}is proportional to Χ_{λ}.

Since *W*(φ_{λ},χ_{λ}) = 1, the Green's function is given by

Direct calculation^{[20]} shows that

where the so-called

Hence

which immediately implies

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

Setting ψ_{λ}^{(0)}=φ_{λ} and ψ_{λ}^{(1)}=χ_{λ}, it follows that

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

The **Mehler–Fock transform**^{[21]}^{[22]}^{[23]} concerns the eigenfunction expansion associated with the Legendre differential operator *D*

on (1,∞). The eigenfunctions are the Legendre functions

with eigenvalue λ ≥ 0. The two Mehler–Fock transformations are

and

(Often this is written in terms of the variable τ = √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,^{[26]} consider the group *G* = SU(1,1) consisting of complex matrices of the form

with determinant |*α*|^{2} − |*β*|^{2} = 1.

A Weyl function can be defined at a singular endpoint giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory.^{[27]} this applies for example to the case of radial Schrödinger operators

The whole theory can also be extended to the case where the coefficients are allowed to be measures.^{[28]}

**^**This is a limit in the strong operator topology.**^**A*bona fide*inner product is defined on the quotient by the subspace of null functions , i.e. those with . Alternatively in this case the support of the measure is , so the right hand side defines a (non-degenerate) inner product on .

**^**Titchmarsh 1962, p. 22**^**Dieudonné 1969, Chapter X.**^**Courant & Hilbert 1989**^**Titchmarsh 1962**^**Titchmarsh, E.C. (1939),*Theory of Functions*, Oxford University Press, §8.2.**^**Burkill, J.C. (1951),*The Lebesgue Integral*, Cambridge Tracts in Mathematics and Mathematical Physics,**40**, Cambridge University Press, pp. 50–52, ISBN 978-0-521-04382-3**^**Loomis, Lynn H. (1953),*An Introduction to Abstract Harmonic Analysis*, van Nostrand, page 40.**^**Loomis 1953, pp. 30–31**^**Kolmogorov, A.N.; Fomin, S.V. (1975),*Introductory Real Analysis*, Dover, pp. 374–376, ISBN 978-0-486-61226-3**^**Riesz & Szőkefalvi-Nagy 1990, p. 263- ^
^{a}^{b}^{c}^{d}^{e}Weyl 1910 .^{[specify]} - ^
^{a}^{b}Bellman 1969, p. 116 **^**Reed & Simon 1975, p. 159**^**Reed & Simon 1975, p. 154**^**Titchmarsh 1946, Chapter III.**^**Kodaira 1949, pp. 935–936**^**Kodaira 1949, pp. 929–932; for omitted details, see Kodaira 1950, pp. 529–536**^**Dieudonné 1988**^**Stone 1932, Chapter X.**^**Kodaira 1950, pp. 534–535**^**Mehler 1881.**^**Fock, V.A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index",*C. R. Acad. Sci. URSS*,**39**: 253–256**^**Vilenkin 1968**^**Terras, Audrey (1984), "Non-Euclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities",*J. Multivariate Anal.*,**15**(2): 261–276, doi:10.1016/0047-259X(84)90031-9**^**Lebedev, N.N. (1972),*Special Functions and Their Applications*, Dover, ISBN 978-0-486-60624-8**^**Vilenkin 1968, Chapter VI.**^**Kostenko, Aleksey; Sakhnovich, Alexander; Teschl, Gerald (2012), "Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials",*Int Math Res Notices*,**2012**: 1699–1747, arXiv:1007.0136, doi:10.1093/imrn/rnr065**^**Eckhardt, Jonathan; Teschl, Gerald (2013), "Sturm–Liouville operators with measure-valued coefficients",*J. d'Analyse Math.*,**120**: 151–224, arXiv:1105.3755, doi:10.1007/s11854-013-0018-x

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