Since ||Tk|| tends to 0, this integral equation has a unique solution given by the Neumann series
f = (I − T)−1h = h + Th + T2h + T3h + ⋯
This iterative scheme is often called Picard iteration after the French mathematician Charles Émile Picard.
If f is twice continuously differentiable (i.e. C2) on (a, b) satisfying Df = λf, 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 C2 eigenfunction fλ on [a, b] with fλ(c) and f 'λ(c) prescribed. Moreover, for each x in [a, b], fλ(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 C2 eigenfunction fλ on (a, b) with fλ(c) and f 'λ(c) prescribed. Moreover, for each x in (a, b), fλ(x) and f 'λ(x) are holomorphic functions of λ.
If f and g are C2 functions on (a, b), the WronskianW(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, gC2 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 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 H0 be the
space of C2 functions f on [a, b] satisfying the Robin boundary conditions
acts on H0. A function f in H0 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 H0, since the Wronskian W(f,g) vanishes if both f,g satisfy the boundary conditions:
It turns out that the eigenvalues can be described by the maximum-minimum principle of Rayleigh–Ritz (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 H0:
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 C1[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.
Green's function (regular case)
From the theory of ordinary differential equations, there are unique fundamental eigenfunctions φλ(x), χλ(x) such that
D φλ = λ φλ, φλ(a) = sin α, φλ'(a) = cos α
D χλ = λ χλ, χλ(b) = sin β, χλ'(b) = cos β
which at each point, together with their first derivatives, depend holomorphically on λ. Let
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).
Gλ(x,y) = φλ (x) χλ(y) / ω(λ) for x ≥ y and χλ(x) φλ (y) / ω(λ) for y ≥ x.
This kernel defines an operator on the inner product space C[a,b] via
Since Gλ(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] = H1 (or equivalently of the dense subspace H0), taking values in H1. This operator carries H1 into H0. When λ is real, Gλ(x,y) = Gλ(y,x) is also real, so defines a self-adjoint operator on H. Moreover,
Gλ (D − λ) =I on H0
Gλ carries H1 into H0, and (D − λ) Gλ = I on H1.
Thus the operator Gλ can be identified with the resolvent (D − λ)−1.
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 ofH0.
In fact let T = Gλ 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 ψn of T with
Tψn = μn ψn, where μn tends to zero. The range of T contains H0 so is dense. Hence 0 is not an eigenvalue of T. The resolvent properties of T imply that ψn lies in H0 and that
D ψn = (λ + 1/μn) ψn
The minimax principle follows because if
then λ(G)= λk 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) ≤ λk.
Wronskian as a Fredholm determinant
For simplicity, suppose that m ≤ q(x) ≤ M on [0,π] with Dirichlet boundary conditions.
The minimax principle shows that
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+(x) and ρ–(x) are the total positive and negative variation of ρ over [a, x].
The support of μ = dρ is the complement of all points x in [a,b] where ρ is constant on some neighborhood of x; by definition it is a closed subset A of [a,b]. Moreover, μ((1-χA)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
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
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 (a, b) requires an initial analysis of the behaviour of the fundamental
eigenfunctions near the endpoints a and b to determine possible boundary conditions there. Unlike the regular Sturm–Liouville case, in some circumstances spectral 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 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 D corresponding to the interval [1,λ]. For an arbitrary function f define
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
into the Banach space E of bounded linear functionals dρ on C[α,β] whenever [α,β] is a compact subinterval of [1, ∞).
Weyl's fundamental observation was that dλf 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 fx(c,λ) may be written as
where ξ1(λ) and ξ2(λ) are given purely in terms of the fundamental eigenfunctions.
The functions of bounded variation
determine a spectral measure on the spectrum of D and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the Titchmarsh–Kodaira formula).
Limit circle and limit point for singular equations
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 c in (0,∞) and, for λ complex, let be the unique fundamental eigenfunctions of D on (0,∞) satisfying
together with the initial conditions at c
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 x > c and by
if x < c.
Let Dx 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 (L2) near 0 or ∞, since lies in Dx for all x>c (in the ∞ case) and so is bounded independent of x. In particular:
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 Dx 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 Dh = λh, so that
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
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
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
on (0,∞) with q0 positive and continuous on (0,∞) and p0 continuously differentiable in [0,∞), positive in (0,∞) and p0(0)=0.
Moreover, assume that after reduction to standard form
D0 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 Φ0 with DΦ0=0 and Φ0 square integrable near 0. In the limit circle case, Φ0 determines a boundary condition at 0:
For λ complex, let Φλ and Χλ satisfy
(D – λ)Φλ = 0, (D – λ)Χλ = 0
Χλ square integrable near infinity
Φλ square integrable at 0 if 0 is limit point
Φλ satisfies the boundary condition above if 0 is limit circle.
a constant which vanishes precisely when Φλ and Χλ 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.
Indeed, if D f= λf with q0 – 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 D0 realization, then for 0 < x < y
Since p0(0) = 0 and f is integrable near 0, p0ff ' must vanish at 0. Setting x = 0, it follows that f(y) f '(y) >0, so that f2 is increasing, contradicting the square integrability of f near ∞.
Thus, adding a positive scalar to q, it may be assumed that
In the examples there will be a third "bad" eigenfunction Ψλ defined and holomorphic for λ not in [1, ∞) such that
Ψλ satisfies the boundary conditions at neither 0 nor ∞. This means that for λ not in [1, ∞)
W(Φλ,Ψλ) is nowhere vanishing;
W(Χλ,Ψλ) is nowhere vanishing.
In this case Χλ is proportional to Φλ + m(λ) Ψλ, where
m(λ) = – W(Φλ,Χλ) / W(Ψλ,Χλ).
Let H1 be the space of square integrable continuous functions on (0,∞) and let H0 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 = G0 by
Then TD = I on H0, DT = I on H1 and the operator D is bounded below on H0:
Thus T is a self-adjoint bounded operator with 0 ≤ T ≤ I.
Formally T = D−1. The corresponding operators Gλ defined for λ not in [1,∞) can be formally identified with
and satisfy Gλ (D – λ) = I on H0, (D – λ)Gλ = I on H1.
Spectral theorem and Titchmarsh–Kodaira formula
Theorem.For every real number λ let ρ(λ) be defined by theTitchmarsh–Kodaira formula:
Then ρ(λ) is a lower semicontinuous non-decreasing function of λ and if
then U defines a unitary transformation of L2(0,∞) onto L2([1,∞), dρ) such that UDU−1corresponds to multiplication by λ.
The inverse transformation U−1 is given by
The spectrum of D equals the support of dρ.
Kodaira gave a streamlined version of Weyl's original proof. (M.H. Stone had previously shown 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 H1 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 E of bounded linear functionals dρ on C[α,β] for any compact subinterval [α,β] of [1, ∞).
The functionals (or measures) dλf(x) satisfies the following E-valued second order ordinary differential equation:
with initial conditions at c in (0,∞)
If φλ and χλ are the special eigenfunctions adapted to c, then
(As the notation suggests, ξλ(0) and ξλ(1) do not depend on the choice of z.)
it follows that
On the other hand, there are holomorphic functions
a(λ), b(λ) such that
φλ + a(λ) χλ is proportional to Φλ;
φλ + b(λ) χλ is proportional to Χλ.
Since W(φλ,χλ) = 1, the Green's function is given by
with eigenvalue λ ≥ 0. The two Mehler–Fock transformations are
(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, 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
A Weyl function can be defined at a singular endpoint giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory. 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.
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