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 SturmâÂÂLiouville 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 TitchmarshâÂÂKodaira 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 SturmâÂÂLiouville 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 SturmâÂÂLiouville 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 MehlerâÂÂFock 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 GelfandâÂÂNaimark 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 be the second order differential operator on given by
where is a strictly positive continuously differentiable function and and are continuous real-valued functions.
For in , define the Liouville transformation by
If
is the unitary operator defined by
then
and
Hence,
where
and
The term in can be removed using an Euler integrating factor. If , then satisfies
where the potential V is given by
The differential operator can thus always be reduced to one of the form
The following is a version of the classical Picard existence theorem for second order differential equations with values in a Banach space .
Let , be arbitrary elements of , a bounded operator on and a continuous function on .
Then, for or , the differential equation
has a unique solution in satisfying the initial conditions
In fact a solution of the differential equation with these initial conditions is equivalent to a solution of the integral equation
with the bounded linear map on defined by
where is the Volterra kernel
and
Since tends to 0, this integral equation has a unique solution given by the Neumann series
This iterative scheme is often called Picard iteration after the French mathematician Charles ÃÂmile Picard.
If is twice continuously differentiable (i.e. ) on satisfying , then is called an eigenfunction of with eigenvalue .
If and are functions on , the Wronskian is defined by
Green's formula - which in this one-dimensional case is a simple integration by parts - states that for , in
When is continuous and , are on the compact interval , this formula also holds for or .
When and are eigenfunctions for the same eigenvalue, then
so that is independent of .
Let be a finite closed interval, a real-valued continuous function on and let be the space of functions on satisfying the Robin boundary conditions
with inner product
In practice usually one of the two standard boundary conditions:
is imposed at each endpoint .
The differential operator given by
acts on . A function in is called an eigenfunction of (for the above choice of boundary values) if for some complex number , the corresponding eigenvalue. By Green's formula, is formally self-adjoint on , since the Wronskian vanishes if both , satisfy the boundary conditions:
As a consequence, exactly as for a self-adjoint matrix in finite dimensions,
It turns out that the eigenvalues can be described by the maximum-minimum principle of RayleighâÂÂRitz (see below). In fact it is easy to see a priori that the eigenvalues are bounded below because the operator is itself bounded below on :
In fact, integrating by parts,
For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with .
For general Robin boundary conditions the first term can be estimated using an elementary Peter-Paul version of Sobolev's inequality:
<blockquote>"Given , there is constant such that for all in ."</blockquote>
In fact, since
only an estimate for is needed and this follows by replacing in the above inequality by for sufficiently large.
From the theory of ordinary differential equations, there are unique fundamental eigenfunctions , such that
which at each point, together with their first derivatives, depend holomorphically on . Let
be an entire holomorphic function.
This function plays the role of the characteristic polynomial of . Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of and, if there are infinitely many, they must tend to infinity. It turns out that the zeros of also have mutilplicity one (see below).
If is not an eigenvalue of on , define the Green's function by
This kernel defines an operator on the inner product space via
Since is continuous on , it defines a HilbertâÂÂSchmidt operator on the Hilbert space completion of (or equivalently of the dense subspace ), taking values in . This operator carries into . When is real, is also real, so defines a self-adjoint operator on . Moreover,
Thus the operator can be identified with the resolvent .
In fact let for large and negative. Then defines a compact self-adjoint operator on the Hilbert space . By the spectral theorem for compact self-adjoint operators, has an orthonormal basis consisting of eigenvectors of with , where tends to zero. The range of contains so is dense. Hence 0 is not an eigenvalue of . The resolvent properties of imply that lies in and that
The minimax principle follows because if
then for the linear span of the first eigenfunctions. For any other -dimensional subspace , some in the linear span of the first eigenvectors must be orthogonal to . Hence .
For simplicity, suppose that on with Dirichlet boundary conditions. The minimax principle shows that
It follows that the resolvent is a trace-class operator whenever is not an eigenvalue of and hence that the Fredholm determinant is defined.
The Dirichlet boundary conditions imply that
Using Picard iteration, Titchmarsh showed that , and hence , is an entire function of finite order :
At a zero of , . Moreover,
satisfies . Thus
This implies that
For otherwise , so that would have to lie in . But then
a contradiction.
On the other hand, the distribution of the zeros of the entire function ÃÂ(û) is already known from the minimax principle.
By the Hadamard factorization theorem, it follows that
for some non-zero constant .
Hence
In particular if 0 is not an eigenvalue of
A function of bounded variation on a closed interval is a complex-valued function such that its total variation , the supremum of the variations
over all dissections
is finite. The real and imaginary parts of are real-valued functions of bounded variation. If is real-valued and normalised so that , it has a canonical decomposition as the difference of two bounded non-decreasing functions:
where and are the total positive and negative variation of over .
If is a continuous function on its RiemannâÂÂStieltjes integral with respect to
is defined to be the limit of approximating sums
as the mesh of the dissection, given by , tends to zero.
This integral satisfies
and thus defines a bounded linear functional on with norm .
Every bounded linear functional on has an absolute value |ü| defined for non-negative by
The form extends linearly to a bounded linear form on with norm and satisfies the characterizing inequality
for in . If is real, i.e. is real-valued on real-valued functions, then
gives a canonical decomposition as a difference of positive forms, i.e. forms that are non-negative on non-negative functions.
Every positive form extends uniquely to the linear span of non-negative bounded lower semicontinuous functions by the formula
where the non-negative continuous functions increase pointwise to .
The same therefore applies to an arbitrary bounded linear form , so that a function of bounded variation may be defined by
where denotes the characteristic function of a subset of . Thus and . Moreover and .
This correspondence between functions of bounded variation and bounded linear forms is a special case of the Riesz representation theorem.
The support of is the complement of all points in where is constant on some neighborhood of ; by definition it is a closed subset of . Moreover, , so that if vanishes on .
Let be a Hilbert space and a self-adjoint bounded operator on 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 .
If is a vector in , then the vectors
form a Cauchy sequence in , since, for ,
and is bounded and increasing, so has a limit.
It follows that can be defined by
If and are vectors in , then
defines a bounded linear form on . By the Riesz representation theorem
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 , 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. Thus extends to a unitary transformation of onto . is then just multiplication by on ; and more generally is multiplication by . In this case, the support of is exactly , so that
The eigenfunction expansion associated with singular differential operators of the form
on an open interval requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints and to determine possible boundary conditions there. Unlike the regular SturmâÂÂLiouville case, in some circumstances spectral values of can have multiplicity 2. In the development outlined below standard assumptions will be imposed on and that guarantee that the spectrum of 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 , for large and positive, is given by an operator corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case was a compact self-adjoint operator; in this case is just a self-adjoint bounded operator with . The abstract theory of spectral measure can therefore be applied to to give the eigenfunction expansion for .
The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of lies in and that and let
be the spectral projection of corresponding to the interval . For an arbitrary function define
may be regarded as a differentiable map into the space of functions of bounded variation ÃÂ; or equivalently as a differentiable map
into the Banach space of bounded linear functionals on whenever is a compact subinterval of .
Weyl's fundamental observation was that satisfies a second order ordinary differential equation taking values in :
After imposing initial conditions on the first two derivatives at a fixed point , 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: and may be written as
where and are given purely in terms of the fundamental eigenfunctions. The functions of bounded variation
determine a spectral measure on the spectrum of and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the TitchmarshâÂÂKodaira formula).
Let be a continuous real-valued function on and let be the second order differential operator
on . Fix a point in and, for complex , let be the unique fundamental eigenfunctions of on satisfying
together with the initial conditions at
Then their Wronskian satisfies
since it is constant and equal to 1 at .
Let be non-real and . 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 and by
if .
Let be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as approaches 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 or , then is square integrable () near or , since lies in for all (in the â case) and so is bounded independent of . In particular:
The radius of the disc can be calculated to be
and this implies that in the limit point case cannot be square integrable near resp. . Therefore, we have a converse to the second statement above:
On the other hand, if for another value , then
satisfies , so that
This formula may also be obtained directly by the variation of constant method from . Using this to estimate , it follows that
More generally if for some function , then
From this it follows that
so that in particular
Similarly
so that in particular
Many more elaborate criteria to be limit point or limit circle can be found in the mathematical literature.
Consider the differential operator
on with positive and continuous on and continuously differentiable in , positive in and .
Moreover, assume that after reduction to standard form becomes the equivalent operator
on where has a finite limit at . Thus
At 0, may be either limit circle or limit point. In either case there is an eigenfunction with and square integrable near . In the limit circle case, determines a boundary condition at :
For complex , let and satisfy
Let
a constant which vanishes precisely when and are proportional, i.e. is an eigenvalue of for these boundary conditions.
On the other hand, this cannot occur if or if is negative.
Indeed, if with , then by Green's formula , since is constant. So must be real. If is taken to be real-valued in the realization, then for
Since and is integrable near , must vanish at . Setting , it follows that , so that is increasing, contradicting the square integrability of near .
Thus, adding a positive scalar to , it may be assumed that
If , the Green's function at is defined by
and is independent of the choice of and .
In the examples there will be a third "bad" eigenfunction defined and holomorphic for not in such that satisfies the boundary conditions at neither nor . This means that for not in
In this case is proportional to , where
Let be the space of square integrable continuous functions on and let be
Define by
Then on , on and the operator is bounded below on :
Thus is a self-adjoint bounded operator with .
Formally . The corresponding operators defined for not in can be formally identified with
and satisfy on , on .
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 with , the spectral projection of is defined by
It is also the spectral projection of corresponding to the interval .
For in define
may be regarded as a differentiable map into the space of functions of bounded variation; or equivalently as a differentiable map
into the Banach space of bounded linear functionals on for any compact subinterval of .
The functionals (or measures) satisfies the following -valued second order ordinary differential equation:
with initial conditions at in
If and are the special eigenfunctions adapted to , then
Moreover,
where
with
(As the notation suggests, and do not depend on the choice of .)
Setting
it follows that
On the other hand, there are holomorphic functions , such that
Since , the Green's function is given by
Direct calculation shows that
where the so-called is given by
Hence
which immediately implies
(This is a special case of the "Stieltjes inversion formula".)
Setting and , it follows that
This identity is equivalent to the spectral theorem and TitchmarshâÂÂKodaira formula.
The MehlerâÂÂFock transform concerns the eigenfunction expansion associated with the Legendre differential operator
on . The eigenfunctions are the Legendre functions
with eigenvalue . The two MehlerâÂÂFock transformations are
and
(Often this is written in terms of the variable .)
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 consisting of complex matrices of the form
with determinant .
A Weyl function can be defined at a singular endpoint giving rise to a singular version of WeylâÂÂTitchmarshâÂÂKodaira 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.