Spectral theory of ordinary differential equations

Template:Short description 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.

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 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.

Solutions of ordinary differential equations

Reduction to standard form

Let Template:Math be the second order differential operator on Template:Open-open given by $$Df(x)=-p(x)f^{″}(x)+r(x)f^{\prime }(x)+q(x)f(x),$$ where Template:Math is a strictly positive continuously differentiable function and Template:Math and Template:Math are continuous real-valued functions.

For Template:Math in Template:Open-open, define the Liouville transformation Template:Math by $$\psi (x)=\underset{x_0}{\overset{x}{\int }}p(t{)}^{-1/2}dt$$

If $$U:L^2(a,b)\mapsto L^2(\psi (a),\psi (b))$$ is the unitary operator defined by $$(Uf)(\psi (x))=f(x)\times {({\psi }^{\prime }(x))}^{-1/2},\mathrm{\forall }x\in (a,b)$$ then $$U\frac{\mathrm{d}}{\mathrm{d}x}{U}^{-1}g=g^{\prime }{\psi }^{\prime }+\frac{1}{2}g\frac{{\psi }^{″}}{{\psi }^{\prime }}$$ and $$\begin{array}{cc}U\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}{U}^{-1}g& =\left(U\frac{\mathrm{d}}{\mathrm{d}x}{U}^{-1}\right)\times \left(U\frac{\mathrm{d}}{\mathrm{d}x}{U}^{-1}\right)g\\ & =\frac{\mathrm{d}}{\mathrm{d}\psi }[g^{\prime }{\psi }^{\prime }+\frac{1}{2}g\frac{{\psi }^{″}}{{\psi }^{\prime }}]\cdot {\psi }^{\prime }+\frac{1}{2}[g^{\prime }{\psi }^{\prime }+\frac{1}{2}g\frac{{\psi }^{″}}{{\psi }^{\prime }}]\cdot \frac{{\psi }^{″}}{{\psi }^{\prime }}\\ & =g^{″}\psi {\text{'}}^2+2g^{\prime }{\psi }^{″}+\frac{1}{2}g\cdot [\frac{{\psi }^{‴}}{{\psi }^{\prime }}-\frac{1}{2}\frac{{\psi }^{\prime }{\text{'}}^2}{\psi {\text{'}}^2}]\end{array}$$

Hence, $$UD{U}^{-1}g=-g^{″}+R{g}^{\prime }+Qg,$$ where $$R=\frac{p^{\prime }+r}{p^{1/2}}$$ and $$Q=q-\frac{r{p}^{\prime }}{4p}+\frac{p^{″}}{4}-\frac{5p{\text{'}}^2}{16p}$$

The term in Template:Math can be removed using an Euler integrating factor. If Template:Math, then Template:Math satisfies $$(SUD{U}^{-1}{S}^{-1})h=-h^{″}+Vh,$$ where the potential V is given by $$V=Q+\frac{S^{″}}{S}$$

The differential operator can thus always be reduced to one of the form $$Df=-f^{″}+qf.$$

Existence theorem

The following is a version of the classical Picard existence theorem for second order differential equations with values in a Banach space Template:Math.

Let Template:Math, Template:Math be arbitrary elements of Template:Math, Template:Math a bounded operator on Template:Math and Template:Math a continuous function on Template:Closed-closed.

Then, for Template:Math or Template:Math, the differential equation $$Df=Af$$ has a unique solution Template:Math in Template:Math satisfying the initial conditions $$f(c)=\beta ,f^{\prime }(c)=\alpha .$$

In fact a solution of the differential equation with these initial conditions is equivalent to a solution of the integral equation $$f=h+Tf$$ with Template:Math the bounded linear map on Template:Math defined by $$Tf(x)=\underset{c}{\overset{x}{\int }}K(x,y)f(y)dy,$$ where Template:Math is the Volterra kernel $$K(x,t)=(x-t)(q(t)-A)$$ and $$h(x)=\alpha (x-c)+\beta .$$

Since Template:Math tends to 0, this integral equation has a unique solution given by the Neumann series $$f=(I-T{)}^{-1}h=h+Th+T^{2}h+T^{3}h+\cdots$$

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

Fundamental eigenfunctions

If Template:Math is twice continuously differentiable (i.e. Template:Math) on Template:Open-open satisfying Template:Math, then Template:Math is called an eigenfunction of Template:Math with eigenvalue Template:Mvar.

• In the case of a compact interval Template:Closed-closed and Template:Math continuous on Template:Closed-closed, the existence theorem implies that for Template:Math or Template:Math and every complex number Template:Mvar there a unique Template:Math eigenfunction Template:Math on Template:Closed-closed with Template:Math and Template:Math prescribed. Moreover, for each Template:Mvar in Template:Closed-closed, Template:Math and Template:Math are holomorphic functions of Template:Mvar.
• For an arbitrary interval Template:Open-open and Template:Math continuous on Template:Open-open, the existence theorem implies that for Template:Math in Template:Open-open and every complex number Template:Mvar there a unique Template:Math eigenfunction Template:Math on Template:Open-open with Template:Math and Template:Math prescribed. Moreover, for each Template:Mvar in Template:Open-open, Template:Math and Template:Math are holomorphic functions of Template:Mvar.

Green's formula

If Template:Math and Template:Math are Template:Math functions on Template:Open-open, the Wronskian Template:Math is defined by $$W(f,g)(x)=f(x)g^{\prime }(x)-f^{\prime }(x)g(x).$$

Green's formula - which in this one-dimensional case is a simple integration by parts - states that for Template:Mvar, Template:Mvar in Template:Open-open $$\underset{x}{\overset{y}{\int }}(Df)g-f(Dg)dt=W(f,g)(y)-W(f,g)(x).$$

When Template:Math is continuous and Template:Math, Template:Math are Template:Math on the compact interval Template:Closed-closed, this formula also holds for Template:Math or Template:Math.

When Template:Math and Template:Math are eigenfunctions for the same eigenvalue, then $$\frac{d}{dx}W(f,g)=0,$$ so that Template:Math is independent of Template:Mvar.

Classical Sturm–Liouville theory

Template:Main Let Template:Closed-closed be a finite closed interval, Template:Math a real-valued continuous function on Template:Closed-closed and let Template:Math be the space of Template:Math functions Template:Math on Template:Closed-closed satisfying the Robin boundary conditions $$\{\begin{array}{c}\cos \alpha f(a)-\sin \alpha f^{\prime }(a)=0,\\ \cos \beta f(b)-\sin \beta f^{\prime }(b)=0,\end{array}$$ with inner product $$(f,g)=\underset{a}{\overset{b}{\int }}f(x)\overline{g(x)}dx.$$

In practice usually one of the two standard boundary conditions:

• Dirichlet boundary condition Template:Math
• Neumann boundary condition Template:Math

is imposed at each endpoint Template:Math.

The differential operator Template:Math given by $$Df=-f^{″}+qf$$ acts on Template:Math. A function Template:Math in Template:Math is called an eigenfunction of Template:Math (for the above choice of boundary values) if Template:Math for some complex number Template:Mvar, the corresponding eigenvalue. By Green's formula, Template:Math is formally self-adjoint on Template:Math, since the Wronskian Template:Math vanishes if both Template:Math, Template:Math satisfy the boundary conditions: $$(Df,g)=(f,Dg),\text{ for }f,g\in H_0.$$

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

• the eigenvalues of Template:Math 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 (see below). In fact it is easy to see a priori that the eigenvalues are bounded below because the operator Template:Math is itself bounded below on Template:Math: Template:Block indent

In fact, integrating by parts, $$(Df,f)={[-f^{\prime }\bar{f}]}_a^b+\int |f^{\prime }{|}^2+\int q|f{|}^2.$$

For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with Template:Math.

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

"Given Template:Math, there is constant Template:Math such that Template:Math for all Template:Math in Template:Math."

In fact, since $$|f(b)-f(x)|\le (b-a{)}^{1/2}\cdot \Vert f^{\prime }{\Vert }_2,$$ only an estimate for Template:Math is needed and this follows by replacing Template:Math in the above inequality by Template:Math for Template:Mvar sufficiently large.

Green's function (regular case)

From the theory of ordinary differential equations, there are unique fundamental eigenfunctions Template:Math, Template:Math such that

• Template:Math, Template:Math, Template:Math
• Template:Math, Template:Math, Template:Math

which at each point, together with their first derivatives, depend holomorphically on Template:Mvar. Let $$\omega (\lambda )=W({\varphi }_{\lambda },{\chi }_{\lambda }),$$ be an entire holomorphic function.

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

If Template:Mvar is not an eigenvalue of Template:Math on Template:Math, define the Green's function by $$G_{\lambda }(x,y)=\{\begin{array}{cc}{\varphi }_{\lambda }(x){\chi }_{\lambda }(y)/\omega (\lambda )& \text{ for }x\ge y\\ {\chi }_{\lambda }(x){\varphi }_{\lambda }(y)/\omega (\lambda )& \text{ for }y\ge x.\end{array}$$

This kernel defines an operator on the inner product space Template:Math via $$(G_{\lambda }f)(x)=\underset{a}{\overset{b}{\int }}{G}_{\lambda }(x,y)f(y)dy.$$

Since Template:Math is continuous on Template:Math, it defines a Hilbert–Schmidt operator on the Hilbert space completion Template:Math of Template:Math (or equivalently of the dense subspace Template:Math), taking values in Template:Math. This operator carries Template:Math into Template:Math. When Template:Mvar is real, Template:Math is also real, so defines a self-adjoint operator on Template:Math. Moreover,

• Template:Math on Template:Math
• Template:Math carries Template:Math into Template:Math, and Template:Math on Template:Math.

Thus the operator Template:Math can be identified with the resolvent Template:Math.

Spectral theorem

In fact let Template:Math for Template:Mvar large and negative. Then Template:Math defines a compact self-adjoint operator on the Hilbert space Template:Math. By the spectral theorem for compact self-adjoint operators, Template:Math has an orthonormal basis consisting of eigenvectors Template:Math of Template:Math with Template:Math, where Template:Math tends to zero. The range of Template:Math contains Template:Math so is dense. Hence 0 is not an eigenvalue of Template:Math. The resolvent properties of Template:Math imply that Template:Math lies in Template:Math and that $$D{\psi }_n=(\lambda +\frac{1}{{\mu }_n}){\psi }_n$$

The minimax principle follows because if $$\lambda (G)=\underset{f\perp G}{\min }\frac{(Df,f)}{(f,f)},$$ then Template:Math for the linear span of the first Template:Math eigenfunctions. For any other Template:Math-dimensional subspace Template:Math, some Template:Math in the linear span of the first Template:Mvar eigenvectors must be orthogonal to Template:Math. Hence Template:Math.

Wronskian as a Fredholm determinant

For simplicity, suppose that Template:Math on Template:Closed-closed with Dirichlet boundary conditions. The minimax principle shows that $$n^2+m\le {\lambda }_n(D)\le n^2+M.$$

It follows that the resolvent Template:Math is a trace-class operator whenever Template:Mvar is not an eigenvalue of Template:Math and hence that the Fredholm determinant Template:Math is defined.

The Dirichlet boundary conditions imply that $$\omega (\lambda )={\varphi }_{\lambda }(b).$$

Using Picard iteration, Titchmarsh showed that Template:Math, and hence Template:Math, is an entire function of finite order Template:Math: $$\omega (\lambda )=𝒪\left(e^{\sqrt{|\lambda |}}\right)$$

At a zero Template:Math of Template:Math, Template:Math. Moreover, $$\psi (x)={\partial }_{\lambda }{\phi }_{\lambda }(x){|}_{\lambda =\mu }$$ satisfies Template:Math. Thus $$\omega (\lambda )=(\lambda -\mu )\psi (b)+𝒪((\lambda -\mu {)}^2)$$

This implies that Template:Block indent

For otherwise Template:Math, so that Template:Math would have to lie in Template:Math. But then $$({\varphi }_{\mu },{\varphi }_{\mu })=((D-\mu )\psi ,{\varphi }_{\mu })=(\psi ,(D-\mu ){\varphi }_{\mu })=0,$$ 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 $$\omega (\lambda )=C\prod (1-\lambda /{\lambda }_n),$$ for some non-zero constant Template:Math.

Hence $$\det (I-\mu (D-\lambda {)}^{-1})=\prod (1-\frac{\mu }{{\lambda }_n-\lambda })=\prod \frac{1-(\lambda +\mu )/{\lambda }_n}{1-\lambda /{\lambda }_n}=\frac{\omega (\lambda +\mu )}{\omega (\lambda )}.$$

In particular if 0 is not an eigenvalue of Template:Math $$\omega (\mu )=\omega (0)\cdot \det (I-\mu D^{-1}).$$

Tools from abstract spectral theory

Functions of bounded variation

Template:See also A function Template:Math of bounded variation on a closed interval Template:Closed-closed is a complex-valued function such that its total variation Template:Math, the supremum of the variations $$\sum _{r=0}^{k-1}|\rho (x_{r+1})-\rho (x_r)|$$ over all dissections $$a=x_0<x_1<\dots <x_k=b$$ is finite. The real and imaginary parts of Template:Math are real-valued functions of bounded variation. If Template:Math is real-valued and normalised so that Template:Math, it has a canonical decomposition as the difference of two bounded non-decreasing functions: $$\rho (x)={\rho }_{+}(x)-{\rho }_{-}(x),$$ where Template:Math and Template:Math are the total positive and negative variation of Template:Math over Template:Closed-closed.

If Template:Math is a continuous function on Template:Closed-closed its Riemann–Stieltjes integral with respect to Template:Math $$\underset{a}{\overset{b}{\int }}f(x)d\rho (x)$$ is defined to be the limit of approximating sums $$\sum _{r=0}^{k-1}f(x_r)(\rho (x_{r+1})-\rho (x_r))$$ as the mesh of the dissection, given by Template:Math, tends to zero.

This integral satisfies $$|\underset{a}{\overset{b}{\int }}f(x)d\rho (x)|\le V(\rho )\cdot \Vert f{\Vert }_{\mathrm{\infty }}$$

and thus defines a bounded linear functional Template:Math on Template:Math with norm Template:Math.

Every bounded linear functional Template:Math on Template:Math has an absolute value |μ| defined for non-negative Template:Math by $$|\mu |(f)=\underset{0\le |g|\le f}{\sup }|\mu (g)|.$$

The form Template:Math extends linearly to a bounded linear form on Template:Math with norm Template:Math and satisfies the characterizing inequality $$|\mu (f)|\le |\mu |(|f|)$$ for Template:Math in Template:Math. If Template:Math is real, i.e. is real-valued on real-valued functions, then $$\mu =|\mu |-(|\mu |-\mu )\equiv {\mu }_{+}-{\mu }_{-}$$ gives a canonical decomposition as a difference of positive forms, i.e. forms that are non-negative on non-negative functions.

Every positive form Template:Math extends uniquely to the linear span of non-negative bounded lower semicontinuous functions Template:Math by the formula $$\mu (g)=\lim \mu (f_n),$$ where the non-negative continuous functions Template:Math increase pointwise to Template:Math.

The same therefore applies to an arbitrary bounded linear form Template:Math, so that a function Template:Math of bounded variation may be defined by $$\rho (x)=\mu ({\chi }_{[a,x]}),$$ where Template:Math denotes the characteristic function of a subset Template:Math of Template:Closed-closed. Thus Template:Math and Template:Math. Moreover Template:Math and Template:Math.

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

The support of Template:Math is the complement of all points Template:Mvar in Template:Closed-closed where Template:Math is constant on some neighborhood of Template:Mvar; by definition it is a closed subset Template:Math of Template:Closed-closed. Moreover, Template:Math, so that Template:Math if Template:Math vanishes on Template:Math.

Spectral measure

Template:See also Let Template:Math be a Hilbert space and $T$ a self-adjoint bounded operator on Template:Math with $0\le T\le I$, so that the spectrum $\sigma (T)$ of $T$ is contained in $[0,1]$. If $p(t)$ is a complex polynomial, then by the spectral mapping theorem $$\sigma (p(T))=p(\sigma (T))$$ and hence $$\Vert p(T)\Vert \le \Vert p{\Vert }_{\mathrm{\infty }}$$ where $\Vert \cdot {\Vert }_{\mathrm{\infty }}$ denotes the uniform norm on Template:Math. By the Weierstrass approximation theorem, polynomials are uniformly dense in Template:Math. It follows that $f(T)$ can be defined $\mathrm{\forall }f\in C[0,1]$, with $$\sigma (f(T))=f(\sigma (T))$$ and $$\Vert f(T)\Vert \le \Vert f{\Vert }_{\mathrm{\infty }}.$$

If $0\le g\le 1$ is a lower semicontinuous function on Template:Closed-closed, for example the characteristic function ${\chi }_{[0,\alpha ]}$ of a subinterval of Template:Closed-closed, then $g$ is a pointwise increasing limit of non-negative $f_n\in C[0,1]$.

If $\xi$ is a vector in Template:Math, then the vectors $${\eta }_n=f_n(T)\xi$$ form a Cauchy sequence in Template:Math, since, for $n\ge m$, $$\Vert {\eta }_n-{\eta }_m{\Vert }^2\le ({\eta }_n,\xi )-({\eta }_m,\xi ),$$ and $({\eta }_n,\xi )=(f_n(T)\xi ,\xi )$ is bounded and increasing, so has a limit.

It follows that $g(T)$ can be defined byTemplate:Efn $$g(T)\xi =\lim f_n(T)\xi .$$

If $\xi$ and Template:Math are vectors in Template:Math, then $${\mu }_{\xi ,\eta }(f)=(f(T)\xi ,\eta )$$ defines a bounded linear form ${\mu }_{\xi ,\eta }$ on Template:Math. By the Riesz representation theorem $${\mu }_{\xi ,\eta }=d{\rho }_{\xi ,\eta }$$ for a unique normalised function ${\rho }_{\xi ,\eta }$ of bounded variation on Template:Closed-closed.

$d{\rho }_{\xi ,\eta }$ (or sometimes slightly incorrectly ${\rho }_{\xi ,\eta }$ itself) is called the spectral measure determined by $\xi$ and Template:Math.

The operator $g(T)$ is accordingly uniquely characterised by the equation $$(g(T)\xi ,\eta )={\mu }_{\xi ,\eta }(g)=\underset{0}{\overset{1}{\int }}g(\lambda )d{\rho }_{\xi ,\eta }(\lambda ).$$

The spectral projection $E(\lambda )$ is defined by $$E(\lambda )={\chi }_{[0,\lambda ]}(T),$$ so that $${\rho }_{\xi ,\eta }(\lambda )=(E(\lambda )\xi ,\eta ).$$

It follows that $$g(T)=\underset{0}{\overset{1}{\int }}g(\lambda )dE(\lambda ),$$ which is understood in the sense that for any vectors $\xi$ and $\eta$, $$(g(T)\xi ,\eta )=\underset{0}{\overset{1}{\int }}g(\lambda )d(E(\lambda )\xi ,\eta )=\underset{0}{\overset{1}{\int }}g(\lambda )d{\rho }_{\xi ,\eta }(\lambda ).$$

For a single vector $\xi ,{\mu }_{\xi }={\mu }_{\xi ,\xi }$ is a positive form on Template:Closed-closed (in other words proportional to a probability measure on Template:Closed-closed) and ${\rho }_{\xi }={\rho }_{\xi ,\xi }$ is non-negative and non-decreasing. Polarisation shows that all the forms ${\mu }_{\xi ,\eta }$ can naturally be expressed in terms of such positive forms, since $${\mu }_{\xi ,\eta }=\frac{1}{4}({\mu }_{\xi +\eta }+i{\mu }_{\xi +i\eta }-{\mu }_{\xi -\eta }-i{\mu }_{\xi -i\eta })$$

If the vector $\xi$ is such that the linear span of the vectors $(T^n\xi )$ is dense in Template:Math, i.e. $\xi$ is a cyclic vector for $T$, then the map $U$ defined by $$U(f)=f(T)\xi ,C[0,1]\to H$$ satisfies $$(U{f}_1,U{f}_2)=\underset{0}{\overset{1}{\int }}{f}_1(\lambda )\overline{f_2(\lambda )}d{\rho }_{\xi }(\lambda ).$$

Let $L_2([0,1],d{\rho }_{\xi })$ denote the Hilbert space completion of $C[0,1]$ associated with the possibly degenerate inner product on the right hand side.Template:Efn Thus $U$ extends to a unitary transformation of $L_2([0,1],{\rho }_{\xi })$ onto Template:Math. $UT{U}^{\ast }$ is then just multiplication by $\lambda$ on $L_2([0,1],d{\rho }_{\xi })$; and more generally $Uf(T)U^{\ast }$ is multiplication by $f(\lambda )$. In this case, the support of $d{\rho }_{\xi }$ is exactly $\sigma (T)$, so that Template:Block indent

Weyl–Titchmarsh–Kodaira theory

The eigenfunction expansion associated with singular differential operators of the form $$Df=-(p{f}^{\prime }{)}^{\prime }+qf$$ on an open interval Template:Open-open requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints Template:Math and Template:Math to determine possible boundary conditions there. Unlike the regular Sturm–Liouville case, in some circumstances spectral values of Template:Math can have multiplicity 2. In the development outlined below standard assumptions will be imposed on Template:Math and Template:Math that guarantee that the spectrum of Template:Math 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 Template:Math, Template:Math for Template:Math large and positive, is given by an operator Template:Math corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case Template:Math was a compact self-adjoint operator; in this case Template:Math is just a self-adjoint bounded operator with Template:Math. The abstract theory of spectral measure can therefore be applied to Template:Math to give the eigenfunction expansion for Template:Math.

The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of Template:Math lies in Template:Closed-open and that Template:Math and let $$E(\lambda )={\chi }_{[{\lambda }^{-1},1]}(T)$$ be the spectral projection of Template:Math corresponding to the interval Template:Closed-closed. For an arbitrary function Template:Math define $$f(x,\lambda )=(E(\lambda )f)(x).$$ Template:Math may be regarded as a differentiable map into the space of functions of bounded variation ρ; or equivalently as a differentiable map $$x\mapsto (d_{\lambda }f)(x)$$ into the Banach space Template:Math of bounded linear functionals Template:Math on Template:Math whenever Template:Closed-closed is a compact subinterval of Template:Closed-open.

Weyl's fundamental observation was that Template:Math satisfies a second order ordinary differential equation taking values in Template:Math: $$D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f.$$

After imposing initial conditions on the first two derivatives at a fixed point Template:Math, this equation can be solved explicitly in terms of the two fundamental eigenfunctions and the "initial value" functionals $$(d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot ),(d_{\lambda }f{)}^{\prime }(c)=d_{\lambda }{f}_x(c,\cdot ).$$

This point of view may now be turned on its head: Template:Math and Template:Math may be written as $$f(c,\lambda )=(f,{\xi }_1(\lambda )),f_x(c,\lambda )=(f,{\xi }_2(\lambda )),$$ where Template:Math and Template:Math 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 Template:Math 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 Template:Math be a continuous real-valued function on Template:Open-open and let Template:Math be the second order differential operator $$Df=-f^{″}+qf$$ on Template:Open-open. Fix a point Template:Math in Template:Open-open and, for complex Template:Mvar, let ${\phi }_{\lambda },{\theta }_{\lambda }$ be the unique fundamental eigenfunctions of Template:Math on Template:Open-open satisfying $$(D-\lambda ){\phi }_{\lambda }=0,(D-\lambda ){\theta }_{\lambda }=0$$ together with the initial conditions at Template:Math $${\phi }_{\lambda }(c)=1,{{\phi }_{\lambda }}^{\prime }(c)=0,{\theta }_{\lambda }(c)=0,{{\theta }_{\lambda }}^{\prime }(c)=1.$$

Then their Wronskian satisfies $$W({\phi }_{\lambda },{\theta }_{\lambda })={\phi }_{\lambda }{{\theta }_{\lambda }}^{\prime }-{\theta }_{\lambda }{{\phi }_{\lambda }}^{\prime }\equiv 1,$$

since it is constant and equal to 1 at Template:Math.

Let Template:Mvar be non-real and Template:Math. If the complex number $\mu$ is such that $f=\phi +\mu \theta$ satisfies the boundary condition $\cos \beta f(x)-\sin \beta f^{\prime }(x)=0$ for some $\beta$ (or, equivalently, $f^{\prime }(x)/f(x)$ is real) then, using integration by parts, one obtains $$\mathrm{Im}(\lambda )\underset{c}{\overset{x}{\int }}|\phi +\mu \theta {|}^2=\mathrm{Im}(\mu ).$$

Therefore, the set of Template:Math satisfying this equation is not empty. This set is a circle in the complex Template:Math-plane. Points Template:Math in its interior are characterized by $$\underset{c}{\overset{x}{\int }}|\phi +\mu \theta {|}^2<\frac{\mathrm{Im}(\mu )}{\mathrm{Im}(\lambda )}$$ if Template:Math and by $$\underset{x}{\overset{c}{\int }}|\phi +\mu \theta {|}^2<\frac{\mathrm{Im}(\mu )}{\mathrm{Im}(\lambda )}$$ if Template:Math.

Let Template:Math be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as Template:Math approaches Template:Math or Template:Math. So in the limit, the circles tend either to a limit circle or a limit point at each end. If $\mu$ is a limit point or a point on the limit circle at Template:Math or Template:Math, then $f=\phi +\mu \theta$ is square integrable (Template:Math) near Template:Math or Template:Math, since $\mu$ lies in Template:Math for all Template:Math (in the ∞ case) and so $\underset{c}{\overset{x}{\int }}|\phi +\mu \theta {|}^2<\frac{\mathrm{Im}(\mu )}{\mathrm{Im}(\lambda )}$ is bounded independent of Template:Mvar. In particular:

• there are always non-zero solutions of Template:Math which are square integrable near Template:Math resp. Template:Math;
• in the limit circle case all solutions of Template:Math are square integrable near Template:Math resp. Template:Math.

The radius of the disc Template:Math can be calculated to be $$\left|\frac{1}{2\mathrm{Im}(\lambda )\underset{c}{\overset{x}{\int }}|\theta {|}^2}\right|$$ and this implies that in the limit point case $\theta$ cannot be square integrable near Template:Math resp. Template:Math. Therefore, we have a converse to the second statement above:

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

On the other hand, if Template:Math for another value Template:Math, then $$h(x)=g(x)-({\lambda }^{\prime }-\lambda )\underset{c}{\overset{x}{\int }}({\phi }_{\lambda }(x){\theta }_{\lambda }(y)-{\theta }_{\lambda }(x){\phi }_{\lambda }(y))g(y)dy$$ satisfies Template:Math, so that $$g(x)=c_1{\phi }_{\lambda }+c_2{\theta }_{\lambda }+({\lambda }^{\prime }-\lambda )\underset{c}{\overset{x}{\int }}({\phi }_{\lambda }(x){\theta }_{\lambda }(y)-{\theta }_{\lambda }(x){\phi }_{\lambda }(y))g(y)dy.$$

This formula may also be obtained directly by the variation of constant method from Template:Math. Using this to estimate Template:Math, it follows that

• the limit point/limit circle behaviour at Template:Math or Template:Math is independent of the choice of Template:Math.

More generally if Template:Math for some function Template:Math, then $$g(x)=c_1{\phi }_{\lambda }+c_2{\theta }_{\lambda }-\underset{c}{\overset{x}{\int }}({\phi }_{\lambda }(x){\theta }_{\lambda }(y)-{\theta }_{\lambda }(x){\phi }_{\lambda }(y))r(y)g(y)dy.$$

From this it follows that

• if Template:Math is continuous at Template:Math, then Template:Math is limit point or limit circle at Template:Math precisely when Template:Math is,

so that in particular

• if Template:Math is continuous at Template:Math, then Template:Math is limit point at Template:Math if and only if Template:Math.

Similarly

• if Template:Math has a finite limit at Template:Math, then Template:Math is limit point or limit circle at Template:Math precisely when Template:Math is,

so that in particular

• if Template:Math has a finite limit at Template:Math, then Template:Math is limit point at Template:Math.

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

Green's function (singular case)

Consider the differential operator $$D_{0}f=-(p_0{f}^{\prime }{)}^{\prime }+q_{0}f$$ on Template:Open-open with Template:Math positive and continuous on Template:Open-open and Template:Math continuously differentiable in Template:Closed-open, positive in Template:Open-open and Template:Math.

Moreover, assume that after reduction to standard form Template:Math becomes the equivalent operator $$Df=-f^{″}+qf$$ on Template:Open-open where Template:Math has a finite limit at Template:Math. Thus

• Template:Math is limit point at Template:Math.

At 0, Template:Math may be either limit circle or limit point. In either case there is an eigenfunction Template:Math with Template:Math and Template:Math square integrable near Template:Math. In the limit circle case, Template:Math determines a boundary condition at Template:Math: $$W(f,{\mathrm{\Phi }}_0)(0)=0.$$

For complex Template:Mvar, let Template:Math and Template:Math satisfy

• Template:Math, Template:Math
• Template:Math square integrable near infinity
• Template:Math square integrable at Template:Math if Template:Math is limit point
• Template:Math satisfies the boundary condition above if Template:Math is limit circle.

Let $$\omega (\lambda )=W({\mathrm{\Phi }}_{\lambda },{\mathrm{X}}_{\lambda }),$$ a constant which vanishes precisely when Template:Math and Template:Math are proportional, i.e. Template:Mvar is an eigenvalue of Template:Math for these boundary conditions.

On the other hand, this cannot occur if Template:Math or if Template:Mvar is negative.

Indeed, if Template:Math with Template:Math, then by Green's formula Template:Math, since Template:Math is constant. So Template:Mvar must be real. If Template:Math is taken to be real-valued in the Template:Math realization, then for Template:Math $$[p_{0}f{f}^{\prime }{]}_x^y=\underset{x}{\overset{y}{\int }}(q_0-\lambda )|f{|}^2+p_0(f^{\prime }{)}^2.$$

Since Template:Math and Template:Math is integrable near Template:Math, Template:Math must vanish at Template:Math. Setting Template:Math, it follows that Template:Math, so that Template:Math is increasing, contradicting the square integrability of Template:Math near Template:Math.

Thus, adding a positive scalar to Template:Math, it may be assumed that $$\omega (\lambda )\ne 0\text{ if }\lambda \notin [1,\mathrm{\infty }).$$

If Template:Math, the Green's function Template:Math at Template:Mvar is defined by $$G_{\lambda }(x,y)=\{\begin{array}{cc}{\mathrm{\Phi }}_{\lambda }(x){\mathrm{X}}_{\lambda }(y)/\omega (\lambda )& (x\le y),\\ {\mathrm{X}}_{\lambda }(x){\mathrm{\Phi }}_{\lambda }(y)/\omega (\lambda )& (x\ge y).\end{array}$$ and is independent of the choice of Template:Math and Template:Math.

In the examples there will be a third "bad" eigenfunction Template:Math defined and holomorphic for Template:Mvar not in Template:Closed-open such that Template:Math satisfies the boundary conditions at neither Template:Math nor Template:Math. This means that for Template:Mvar not in Template:Closed-open

• Template:Math is nowhere vanishing;
• Template:Math is nowhere vanishing.

In this case Template:Math is proportional to Template:Math, where $$m(\lambda )=-W({\mathrm{\Phi }}_{\lambda },{\mathrm{X}}_{\lambda })/W({\mathrm{\Psi }}_{\lambda },{\mathrm{X}}_{\lambda }).$$

Let Template:Math be the space of square integrable continuous functions on Template:Open-open and let Template:Math be

• the space of Template:Math functions Template:Math on Template:Open-open of compact support if Template:Math is limit point at Template:Math
• the space of Template:Math functions Template:Math on Template:Open-open with Template:Math at Template:Math and with Template:Math near Template:Math if Template:Math is limit circle at Template:Math.

Define Template:Math by $$(Tf)(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{G}_0(x,y)f(y)dy.$$

Then Template:Math on Template:Math, Template:Math on Template:Math and the operator Template:Math is bounded below on Template:Math: $$(Df,f)\ge (f,f).$$

Thus Template:Math is a self-adjoint bounded operator with Template:Math.

Formally Template:Math. The corresponding operators Template:Math defined for Template:Mvar not in Template:Closed-open can be formally identified with $$(D-\lambda {)}^{-1}=T(I-\lambda T{)}^{-1}$$ and satisfy Template:Math on Template:Math, Template:Math on Template:Math.

Spectral theorem and Titchmarsh–Kodaira formula

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 Template:Math with Template:Math, the spectral projection Template:Math of Template:Math is defined by $$E(\lambda )={\chi }_{[{\lambda }^{-1},1]}(T)$$

It is also the spectral projection of Template:Math corresponding to the interval Template:Closed-closed.

For Template:Math in Template:Math define $$f(x,\lambda )=(E(\lambda )f)(x).$$

Template:Math may be regarded as a differentiable map into the space of functions Template:Math of bounded variation; or equivalently as a differentiable map $$x\mapsto (d_{\lambda }f)(x)$$ into the Banach space Template:Math of bounded linear functionals Template:Math on Template:Closed-closed for any compact subinterval Template:Closed-closed of Template:Closed-open.

The functionals (or measures) Template:Math satisfies the following Template:Math-valued second order ordinary differential equation: $$D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f,$$ with initial conditions at Template:Math in Template:Open-open $$(d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot )={\mu }^{(0)},(d_{\lambda }f{)}^{\prime }(c)=d_{\lambda }{f}_x(c,\cdot )={\mu }^{(1)}.$$

If Template:Math and Template:Math are the special eigenfunctions adapted to Template:Math, then $$d_{\lambda }f(x)={\phi }_{\lambda }(x){\mu }^{(0)}+{\chi }_{\lambda }(x){\mu }^{(1)}.$$

Moreover, $${\mu }^{(k)}=d_{\lambda }(f,{\xi }_{\lambda }^{(k)}),$$ where $${\xi }_{\lambda }^{(k)}=DE(\lambda ){\eta }^{(k)},$$ with $${\eta }_z^{(0)}(y)=G_z(c,y),{\eta }_z^{(1)}(x)={\partial }_x{G}_z(c,y),(z\notin [1,\mathrm{\infty })).$$ (As the notation suggests, Template:Math and Template:Math do not depend on the choice of Template:Mvar.)

Setting $${\sigma }_{ij}(\lambda )=({\xi }_{\lambda }^{(i)},{\xi }_{\lambda }^{(j)}),$$ 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 Template:Math, Template:Math such that

• Template:Math is proportional to Template:Math;
• Template:Math is proportional to Template:Math.

Since Template:Math, the Green's function is given by $$G_{\lambda }(x,y)=\{\begin{array}{cc}\frac{({\phi }_{\lambda }(x)+a(\lambda ){\chi }_{\lambda }(x))({\phi }_{\lambda }(y)+b(\lambda ){\chi }_{\lambda }(y))}{b(\lambda )-a(\lambda )}& (x\le y),\\ \frac{({\phi }_{\lambda }(x)+b(\lambda ){\chi }_{\lambda }(x))({\phi }_{\lambda }(y)+a(\lambda ){\chi }_{\lambda }(y))}{b(\lambda )-a(\lambda )}& (y\le x).\end{array}$$

Direct calculation shows that $$({\eta }_z^{(i)},{\eta }_z^{(j)})=\mathrm{Im}{M}_{ij}(z)/\mathrm{Im}z,$$ where the so-called Template:Em Template:Math is given by $$M_{00}(z)=\frac{a(z)b(z)}{a(z)-b(z)},M_{01}(z)=M_{10}(z)=\frac{a(z)+b(z)}{2(a(z)-b(z))},M_{11}(z)=\frac{1}{a(z)-b(z)}.$$

Hence $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\mathrm{Im}z)\cdot |\lambda -z{|}^{-2}d{\sigma }_{ij}(\lambda )=\mathrm{Im}{M}_{ij}(z),$$ which immediately implies $${\sigma }_{ij}(\lambda )=\underset{\delta \downarrow 0}{\lim }\underset{\epsilon \downarrow 0}{\lim }\underset{\delta }{\overset{\lambda +\delta }{\int }}\mathrm{Im}{M}_{ij}(t+i\epsilon )dt.$$ (This is a special case of the "Stieltjes inversion formula".)

Setting Template:Math and Template:Math, it follows that $$(E(\mu )f)(x)=\sum _{i.j}\underset{0}{\overset{\mu }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{\psi }_{\lambda }^{(i)}(x){\psi }_{\lambda }^{(j)}(y)f(y)dyd{\sigma }_{ij}(\lambda )=\underset{0}{\overset{\mu }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Phi }}_{\lambda }(x){\mathrm{\Phi }}_{\lambda }(y)f(y)dyd\rho (\lambda ).$$

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

Application to the hypergeometric equation

Template:See also The Mehler–Fock transformTemplate:Sfn concerns the eigenfunction expansion associated with the Legendre differential operator Template:Math $$Df=-((x^2-1)f^{\prime }{)}^{\prime }=-(x^2-1)f^{″}-2x{f}^{\prime }$$ on Template:Open-open. The eigenfunctions are the Legendre functions $$P_{-1/2+i\sqrt{\lambda }}(\cosh r)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{\left(\frac{\sin \theta +i{e}^{-r}\cos \theta }{\cos \theta -i{e}^{-r}\sin \theta }\right)}^{\frac{1}{2}+i\sqrt{\lambda }}d\theta$$ with eigenvalue Template:Math. The two Mehler–Fock transformations are $$Uf(\lambda )=\underset{1}{\overset{\mathrm{\infty }}{\int }}f(x)P_{-1/2+i\sqrt{\lambda }}(x)dx$$ and $$U^{-1}g(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}g(\lambda )\frac{1}{2}\tanh \pi \sqrt{\lambda }d\lambda .$$

(Often this is written in terms of the variable Template:Math.)

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 Template:Math consisting of complex matrices of the form $$\left[\begin{array}{cc}\alpha & \beta \\ \bar{\beta }& \bar{\alpha }\end{array}\right]$$

with determinant Template:Math.

Generalisations and alternative approaches

A Weyl function can be defined at a singular endpoint Template:Math giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory. this applies for example to the case of radial Schrödinger operators $$Df=-f^{″}+\frac{\ell (\ell +1)}{x^2}f+V(x)f,x\in (0,\mathrm{\infty })$$

The whole theory can also be extended to the case where the coefficients are allowed to be measures.

References

Bibliography

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