Spectral theory of ordinary differential equations

In mathematics, the spectral theory of ordinary differential equations is 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.

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 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 resolvant of the singular differential operator could be approximated by compact resolvants 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 I. M. 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 F. G. 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.

Solutions of ordinary differential equations

Reduction to standard form

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

:$Df(x)\; =\; -p(x)\; f^\{primeprime\}(x)\; +r(x)\; f^prime(x)\; +\; q(x)\; f(x),$

where "p" is a strictly positive continuously differentiable function and "q" and "r" are continuousreal-valued functions.

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

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

Let "U" be the unitary operator("Uf")·ψ ψ'^½ = "f" from "L"² ("a","b") onto"L"²("c","d"), where "c" = ψ("a") and "d" = ψ("b"). Then :$UDU^\{-1\}\; g=\; -g^\{primeprime\}\; +\; R\; g^\{prime\}\; +\; Q\; g,$

where "Q"·ψ = "q" and "R"·ψ= ("p' "/2 + "r")/"p"^½.The term in "g' " can be removed using an Euler integrating factor. If "S' "/"S" = -"R"/2, then "h" = "Sg"satisfies :$(S\; UDU^\{-1\}\; S^\{-1\})\; h\; =\; -h^\{prime\; prime\}\; +\; V\; h,$

where the potential "V" is given by "V" = "Q" + "S"²/2 + "S' "/2.The differential operator can thus always be reduced to one of the form [ Titchmarsh (1962), Page 22. ]

:$Df\; =\; -\; f^\{primeprime\}\; +\; qf.$

Existence theorem

The following is a version of the classical Picard existence theorem for second order differential equations with values in a
Banach space E. [Dieudonné (1969), Chapter X. ]

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 solutionof the integral equation

:"f" = "h" + "T" "f"

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

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

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.

Fundamental eigenfunctions

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.

Green's formula

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 states that for "x", "y" in ("a", "b")

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

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

:$\{dover\; dx\}\; W(f,g)\; =0,$

so that "W"("f", "g") is independent of "x".

Classical Sturm-Liouville theory

Let ["a", "b"] be a finite closed interval, "q" a real-valued continuous function on ["a", "b"] and let "H"₀ be thespace of C² functions "f" on ["a", "b"] satisfying the mixed boundary conditions

:

with inner product

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

In practise usually one of the two unmixed boundary conditions:

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

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

The differential operator "D" given by

:$Df=-f^\{primeprime\}\; +\; qf$

acts on "H"₀. A function "f" in "H"₀ is called an eigenfunction of "D" (for the above choice of boundary values) if "Df" = λ "f" for some complex number λ, the corresponding eigenvalue.By Green's formula, "D" is formally self-adjoint on "H"₀:

:("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 [ Courant & Hilbert (1989). ] (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:

:*$(Df,\; f)\; ge\; M\; (f,\; f)$ for some finite constant $M$.

In fact integrating by parts

::$(Df,f)=\; [-f^prime\; overline\{f\}]\; \_a^b\; +\; int\; |f^prime|^2\; +\; int\; q\; |f|^2.$

For unmixed boundary conditions, the first term vanishes and the inequality holds with "M" = inf "q".

In the mixed case 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 C¹ [a, b] ."

In fact, since

::|"f"("b") - "f"("x")| ≤ ("b" - "a")^½·||"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.

Green's function (regular case)

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),

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

:$(G\_lambda\; f)(x)\; =int\_a^b\; G\_lambda(x,y)\; f(y),\; dy.$

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 resolvant ("D" - λ)^-1.

Spectral theorem

Theorem. "The eigenvalues of D are real of multiplicity one and form an increasing sequence λ₁ < λ₂ < ··· 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"

:$lambda\_k\; =\; max\_,\; pages\; 374-376.]$

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

where χ_"A" denotes the characteristic function of a subset "A" of ["a", "b"] . Thus μ = "d"ρ and ||μ|| = ||"d"ρ||.Moreover μ₊ = "d"ρ₊ and μ_– = "d"ρ_–.

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 neigbourhood 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".

pectral measure

Let "H" be a Hilbert space and "T" a self-adjoint bounded operator on "H" with 0 ≤ "T" ≤ I, so that the spectrum σ("T") of "T" is contained in [0,1] . If "p"("t") is a complex polynomial, then by the "spectral mapping theorem"

:σ("p"("T"))= "p"(σ("T"))

and hence

:||"p"("T")|| ≤ ||"p"||_∞,

where ||·||_∞ denotes the uniform norm on C [0,1] . By the Weierstrass approximation theorem, polynomials are uniformly dense in C [0, 1] . It follows that "f"("T") can be defined for every "f" in C [0,1] , with

:σ("f"("T"))= "p"(f("T")) and ||"f"("T")|| ≤ ||"f"||_∞.

If 0 ≤ "g" ≤ 1 is a lower semicontinuous function on [0,1] , for example the characteristic function χ _[0,α] of a subinterval of [0,1] , then"g" is a pointwise increasing limit of non-negative "f"_"n" in C [0,1] .

According to Sz.-Nagy, [Riesz & Nagy (1990), page 263.] if ξ is a vector in "H", then the vectors

:η_"n" = "f"_"n"("T") ξ

form a Cauchy sequence in "H", since, for "n" ≥ "m",

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

and (η_"n",ξ) = ("f"_"n"(T)ξ, ξ) is bounded and increasing, so has a limit.

It follows that "g"("T") can be defined by [This is a limit in the strong operator topology.]

:"g"("T")ξ = lim "f"_"n"("T")ξ.

If ξ and η are vectors in "H", then

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

defines a bounded linear form μ_ξ,η on "H". By the Riesz representation theorem

:$mu\_\{xi,eta\}=d\; ho\_\{xi,eta\}$

for a unique normalised function ρ_ξ,η of bounded variation on [0,1] .

dρ_ξ,η (or sometimes slightly incorrectly ρ_ξ,η itself) is called the spectral measuredetermined by ξ and η. The operator "g"("T") is accordingly uniquely characterised by the equation

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

The spectral projection "E"(λ) is defined by

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

so that

:$ho\_\{xi,eta\}(lambda)=(E(lambda)xi,eta).$

It follows that

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

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

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

For a single vector ξ, μ_ξ = μ_ξ,ξ is a positive form on [0,1] (in other words proportional to a probability measure on [0,1] ) and ρ_ξ = ρ_ξ,ξ is non-negative and non-decreasing.Polarisation shows that all the forms μ_ξ,η can naturally be expressed in terms of such positive forms, since

:μ_ξ,η = ¼(μ_ξ+η + i μ_ξ+iη - μ_ξ-η -i μ_ξ-iη).

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

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

satisfies

:$(Uf\_1,Uf\_2)=\; int\_0^1\; f\_1(lambda)\; overline\{f\_2(lambda)\}\; ,\; d\; ho\_xi(lambda).$

Let L²( [0,1] , "d"ρ_ξ) denote the Hilbert space completion of C [0,1] associatedwith the possibly degenerate inner product on the right hand side. [A "bona fide" inner product is defined on the quotient by the subspace of null functions "f", i.e. those with μ_ξ(|"f"|²)=0. Alternatively in this case the support of the measure is σ("T"), so the right hand side defines a (non-degenerate) inner product on C(σ("T")).] Thus "U" extends to a unitary transformation of L²( [0,1] , "d"ρ_ξ) onto "H". "UTU"* is then just multiplication by λ on L²( [0,1] , "d"ρ_ξ); and more generally "Uf(T)U"* is multiplication by "f"(λ). In this case, the support of "d"ρ_ξis exactly σ("T"), so that

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

Weyl-Kodaira theory

The eigenfunction expansion associated with singular differential operators of the form

:$Df\; =\; -(pf^prime)^prime\; +\; qf$

on an open interval ("a", "b") requires an initial analysis of the behaviour of the fundamentaleigenfunctions near the endpoints "a" and "b" to determine possible boundary conditions there. Unlike the regular Sturm-Liouville case, in some circumstances spectral values of "D" can have multiplicity 2. In the development outlined below standard assumptions will be imposed on "p" and "q" that guarantee that the spectrum of"D" has multiplicity one everywhere and is bounded below. This includes almost all important applications; modifications required for the more general case will be discussed later.

Having chosen the boundary conditions, as in the classical theory the resolvant of "D", ("D" + "R" )^-1 for "R " large and positive, isgiven 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

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

be the spectral projection of "D" corresponding to the interval [1,λ] . For an arbitrary function "f" define

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

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

:$xmapsto\; (d\_lambda\; f)(x)$

into the Banach space E of bounded linear functionals "d"ρ on C [α,β] whenever [α,β] is a compact subinterval of [1, ∞).

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

:$D\; (d\_lambda\; f)\; =\; lambda\; cdot\; d\_lambda\; f.$

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

:$(d\_lambda\; f)(c)=\; d\_lambda\; f(c,cdot),\; quad\; (d\_lambda\; f)^prime(c)=\; d\_lambda\; f\_x(c,cdot).$

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

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

where ξ₁(λ) and ξ₂(λ) are given purely in terms of the fundamental eigenfunctions.The functions of bounded variation

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

determine a spectral measure on the spectrum of "D" and can be computed explicitly from the behaviourof the fundamental eigenfunctions (the Titchmarsh-Kodaira formula).

Limit circle and limit point for singular equations

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

:$Df=\; -f^\{primeprime\}\; +\; qf$

on (0,∞). Fix a point "c" in (0,∞) and, for λ complex, let φ_λ, χ_λ be the unique fundamental eigenfunctions of "D" on (0,∞) satisfying

:$(D-lambda)varphi\_lambda\; =\; 0,\; quad\; (D-lambda)chi\_lambda\; =0$

together with the initial conditions at "c"

:$varphi\_lambda(c)=1,,\; varphi\_lambda^prime(c)=0,\; ,\; chi\_lambda(c)=0,\; ,\; chi\_lambda^prime(c)=1.$

Then their Wronskian satisfies

:$W(varphi\_lambda,chi\_lambda)\; =\; varphi\_lambdachi\_lambda^prime-\; chi\_lambda\; varphi\_lambda^primeequiv\; 1,$

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

If λ is non-real and 0 < "x" < ∞, then Green's formula implies

:$2\; ,\; \{\; m\; Im\}(lambda)\; int\_c^x\; |varphi\; +mu\; chi|^2\; =-\{\; m\; Im\}(mu).$

This defines a circle in the complex μ-plane, the interior of which is given by

: m Im}(mu)over 2, { m Im}(lambda)}

if "x" > "c" and by

: m Im}(mu)over 2, { m Im}(lambda)}

if "x" < "c".

Let "D"_"x" be the closed disc enclosed by the circle. By definitionthese closed discs are nested and decrease as "x" approaches 0 or ∞. So in the limit, the circlestend either to a limit circle or a limit point at each end. In particular if μ is a limit point or a point on the limit circle at 0 or ∞, then |φ + μχ|² is
square integrable (L²) near 0 or ∞. In particular: [ Weyl (1910), Math. Ann. ]

* "there are always non-zero solutions of Df = λf which are square integrable near 0 or ∞";
* "there is exactly one non-zero solution (up to scalar multiples) of Df = λf which is square integrable near 0 or ∞ precisely in the limit point case".

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

:$h(x)\; =\; g(x)\; -(lambda^prime-lambda)\; int\_c^x\; (varphi\_lambda(x)\; chi\_lambda(y)\; -\; chi\_lambda(x)varphi\_lambda(y))g(y),\; dy$

satisfies "Dh" = λ"h", so that

:$g(x)=c\_1\; varphi\_lambda\; +\; c\_2\; chi\_lambda\; +\; (lambda^prime-lambda)\; int\_c^x\; (varphi\_lambda(x)\; chi\_lambda(y)\; -\; chi\_lambda(x)varphi\_lambda(y))g(y),\; dy.$

Using this to estimate "g", it follows that [ Weyl (1910), Math. Ann. ]

* "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 [ Bellman (1969), page 116.]

:$g(x)=c\_1\; varphi\_lambda\; +\; c\_2\; chi\_lambda\; -\; int\_c^x\; (varphi\_lambda(x)\; chi\_lambda(y)\; -\; chi\_lambda(x)varphi\_lambda(y))r(y)g(y),\; dy.$

From this it follows that [ Bellman (1969), page 116.]

* "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 [Reed and Simon (1975), page 159. ]

* "if q(x)- a/x² 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 [Reed and Simon (1975), page 154. ]

* "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.

Green's function (singular case)

Consider the differential operator

:$D\_0\; f\; =\; -(p\_0f^prime)^prime\; +\; q\_0f$

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

:$Df=\; -f^\{primeprime\}\; +\; qf$

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 Φ₀ with "D"Φ₀=0 and Φ₀ square integrable near 0. In the limit circle case, Φ₀ determines a boundary condition at 0:

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

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

:$omega(lambda)\; =\; W(Phi\_lambda,Chi\_lambda),$

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

On the other hand, this cannot occur if Im λ ≠ 0 or if λ is negative. [ Weyl (1910), Math. Ann. ]

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"

:$[p\_0\; ff^prime]\; \_x^y\; =\; int\_x^y\; (q\_0\; -lambda)|f|^2\; +\; p\_0\; (f^prime)^2\; .$

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

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

:$G\_lambda(x,y)\; =\; Phi\_lambda(x)Chi\_lambda(y)/omega(lambda)\; ,,\; (xle\; y),\; ,,,,\; Chi\_lambda(x)Phi\_lambda(y)/omega(lambda)\; ,,\; (xge\; y).$

and is independent of the choice of _λ and Χ_λ.

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 C² functions "f" on (0,∞) of compact support if "D" is limit point at 0
* the space of C² functions "f" on (0,∞) with "W"("f",Φ₀)=0 at 0 and with "f" = 0 near ∞ if "D" is limit circle at 0.

Define "T" = "G"₀ by

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

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

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

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

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

and satisfy "G"_λ ("D" – λ) = "I" on "H"₀, ("D" – λ)"G"_λ = "I" on "H"₁.

Spectral theorem and Titchmarsh-Kodaira formula

Theorem. [ Weyl (1910), Math. Ann.] [Titchmarsh (1946), Chapter III.] [ Kodaira (1949), pages 935-936.] "For every real number λ let ρ(λ) be defined by the" Titchmarsh-Kodaira formula:ψψ:$ho(lambda)\; =\; lim\_\{delta\; downarrow\; 0\}\; lim\_\{varepsilon\; downarrow\; 0\}\; \{1over\; pi\}\; int\_delta^\{lambda+delta\}\; \{\; m\; Im\},\; m(t\; +\; ivarepsilon)\; ,\; dt.$

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

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

"then U defines a unitary transformation of L²(0,∞) onto L²( [1,∞), dρ) such that"UDU^–1 "corresponds to multiplication by λ. "

"The inverse transformation U^–1 is given by"

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

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

Kodaira gave a streamlined version [ Kodaira (1949), 929-932; for omitted details, see Kodaira (1950), 529-536.] [ Dieudonné (1988).] of Weyl's original proof. [Weyl (1910), Math. Ann.] (M.H. Stone had previously shown [ Stone (1932), Chapter X.] 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

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

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

For "f" in "H"₁ define

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

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

:$xmapsto\; (d\_lambda\; f)(x)$

into the Banach space E of bounded linear functionals "d"ρ on C [α,β] for any compact subinterval [α,β] of [1, ∞).

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

:$D\; (d\_lambda\; f)\; =\; lambda\; cdot\; d\_lambda\; f,$

with initial conditions at "c" in (0,∞)

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

If φ_λ and χ_λ are the special eigenfunctions adapted to "c", then

:$d\_lambda\; f\; (x)\; =\; varphi\_lambda(x)\; mu^\{(0)\}\; +\; chi\_lambda(x)\; mu^\{(1)\}.$

Moreover

:$mu^\{(k)\}=\; d\_lambda\; (f,xi^\{(k)\}\_lambda),$

where

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

with

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

(As the notation suggests, ξ_λ⁽⁰⁾ and ξ_λ⁽¹⁾ do not depend on the choice of "z".)

Setting

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

it follows that

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

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

* φ_λ + "a"(λ) χ_λ is proportional to Φ_λ;
* φ_λ + "b"(λ) χ_λ is proportional to Χ_λ.

Since "W"(φ_λ,χ_λ) = 1, the Green's function is given by

:$G\_lambda(x,y)\; =\; \{(varphi\_lambda(x)\; +\; a(lambda)chi\_lambda(x))(varphi\_lambda(y)\; +\; b(lambda)chi\_lambda(y))over\; b(lambda)-a(lambda)\}\; ,,\; (xle\; y),\; ,,,,\; \{(varphi\_lambda(x)\; +\; b(lambda)chi\_lambda(x))(varphi\_lambda(y)\; +\; a(lambda)chi\_lambda(y))over\; b(lambda)-a(lambda)\}\; ,,\; (yle\; x).$

Direct calculation [Kodaira (1950), pages 534-535.] shows that

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

where the so-called "characteristic matrix" "M"_"ij"("z") is given by

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

Hence

:$int\_\{-infty\}^infty\; (\{\; m\; Im\},\; z)cdot|lambda-z|^\{-2\},\; dsigma\_\{ij\}(lambda)\; =\; \{\; m\; Im\}\; M\_\{ij\}(z),$

which immediately implies

:$sigma\_\{ij\}(lambda)\; =\; lim\_\{deltadownarrow\; 0\}\; lim\_\{varepsilon\; downarrow\; 0\}\; int\_delta^\{lambda\; +delta\}\{\; m\; Im\},\; M\_\{ij\}(t\; +ivarepsilon),\; dt.$

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

Setting ψ_λ⁽⁰⁾=φ_λ and ψ_λ⁽¹⁾=χ_λ, it follows that

:$(E(mu)f)(x)=\; sum\_\{i.j\}int\_0^mu\; int\_0^inftypsi^\{(i)\}\_lambda(x)psi^\{(j)\}\_lambda(y)\; f(y),\; dy\; ,dsigma\_\{ij\}(lambda)\; =\; int\_0^mu\; int\_0^inftyPhi\_lambda(x)\; Phi\_lambda(y)\; f(y),\; dy\; ,\; d\; ho(lambda).$

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

Application to the hypergeometric equation

The Mehler-Fock transform [citation|last=Mehler|first=F.G.|title=Ueber mit der Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorieder Elektricitätsverteilung|journal=Math. Annalen|year=1881|volume=18|pages=161-194] [citation|last=Fock|first=V.A.|title=On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index|journal=C. R. (Doklady) Acad. Sci. URSS |year=1943|volume=39|pages=253-256] [ Vilenkin (1968).] concerns the eigenfunction expansion associated with the Legendre differential operator "D"

:$Df\; =\; -((x^2-1)\; f^prime)^prime\; =-(x^2-1)f^\{primeprime\}\; -2x\; f^prime$

on (1,∞). The eigenfunctions are the Legendre functions [citation|first=Audrey|last=Terras|title=Non-Euclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities|journal=J. Multivariate Anal.|volume= 15 |year=1984|pages=261-276]

:$P\_\{-1/2+isqrt\{lambda$(cosh r) = {1over 2pi} int_0^{2pi} left({sin heta + i e^{-r} cos hetaover cos heta - i e^{-r}sin heta} ight)^1over 2}+isqrt{lambda, d heta

with eigenvalue λ ≥ 0. The two Mehler-Fock transformations are [citation|first=N.N.|last=Lebedev|title=Special Functions and Their Applications|year=1972
publisher=Dover|id=ISBN 0486606244]

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

and

:$U^\{-1\}g(x)=int\_0^infty\; g(lambda)\; ,\; \{1over2\}\; anh\; pi\; sqrt\{lambda\},dlambda.$

(Often this is written in terms of the variable 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, [ Vilenkin (1968), Chapter VI.] consider the group "G" = SU(1,1) consisting of complex matrices of the form

:

with determinant |α|² - |β|² = 1.

Application to the hydrogen atom

Generalisations and alternative approaches

Gelfand-Levitan theory

Notes

References

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