Titchmarsh-Weyl m-function

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A function arising in an attempt to properly determine which singular boundary-value problems are self-adjoint (cf. also Self-adjoint differential equation). Begin with a formally symmetric differential expression

$$ L y = \frac{-(p y')' + q y}{w} , $$

where $p\ne 0$, $q,w>0$ are measurable coefficients over $[a,b)$, and which is defined on a domain within $L^2(a,b;w)$. The Titchmarsh–Weyl $m$-function is defined as follows: For $\lambda = \mu + i \nu$, $\nu\ne 0$, let $\phi$ and $\psi$ be solutions of $L y = \lambda y$ satisfying

$$ \begin{aligned} \phi(a,\lambda) &= \sin\alpha, & \psi(a,\lambda) &= \cos\alpha, \\ p\phi'(a,\lambda) &= -\cos\alpha, & p\psi'(a,\lambda) &= \sin\alpha . \end{aligned} $$

Now consider a real boundary condition at $b'$, $a<b'<b$, of the form

$$\cos\beta\, x(b')+\sin\beta\, px'(b')=0,$$

and let $\chi(x,\lambda)=\phi(x,\lambda)+\ell(\lambda)\psi(x,\lambda)$ satisfy it. Then


If $z=\cot\beta$, $\ell$ is a meromorphic function in the complex $z$-plane; indeed, it is a fractional-linear transformation of the $z$-plane into itself. From the well-known properties of fractional-linear transformations, as $\beta$ varies over real values $0\leq\beta\leq\pi$, $z$ varies over the real $z$-axis, and $\ell$ describes a circle in the $z$-plane.

It can be shown that if $b'$ increases, the circles become nested. Hence there is at least one point inside all. For such a point $\ell=m(\lambda)$,


There exists at least one solution of $Ly=\lambda y$, which is square-integrable.

If the limit of the circles is a point, then $m(\lambda)$ is unique and only $\chi(x,\lambda)$ is square-integrable. This is the limit-point case. If the limit of the circles is itself a circle, then $m(\lambda)$ is not unique and all solutions of $Ly=\lambda y$ are square-integrable. This is the limit-circle case.

Nonetheless, the differential operator


whose domain satisfies

$$\sin\alpha\, y(a)-\cos\alpha\, py'(a)=0,$$

$$\lim_{x\to b}[p(x)(y(x)\chi'(\lambda,x)-y'(x)\chi(x,\lambda)]=0,$$

where $\ell=m$ on the limit circle or limit point, is a self-adjoint differential operator (cf. also Self-adjoint operator; Self-adjoint differential equation) on $L^2(a,b;w)$.

If the circle limit is a point, the second boundary condition (at $b$) is automatic.

The spectral measure of $L$ is given by


The spectral resolution of arbitrary functions in $L^2(a,b;w)$ is

$$f(x) = \lim_{(\mu,\nu) \to (-\infty,\infty)} \int_\mu^\nu g(\lambda) \psi(x,\lambda) d\rho(\lambda) , $$

where the limit is in the mean-square sense, and

$$g(\lambda)=\lim_{b'\to b}\int\limits_a^{b'}f(x)\psi(x,\lambda)dx.$$


[a1] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[a2] A.M. Krall, "$M(\lambda)$ theory for singular Hamiltonian systems with one singular point" SIAM J. Math. Anal. , 20 (1989) pp. 644–700
How to Cite This Entry:
Titchmarsh-Weyl m-function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Allan M. Krall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article