Titchmarsh-Weyl m-function

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 , , of the form

and let satisfy it. Then

If , is a meromorphic function in the complex -plane; indeed, it is a bilinear transformation. As varies over real values , varies over the real -axis, and describes a circle in the -plane.

It can be shown that if increases, the circles become nested. Hence there is at least one point inside all. For such a point ,

There exists at least one solution of , which is square-integrable.

If the limit of the circles is a point, then is unique and only is square-integrable. This is the limit-point case. If the limit of the circles is itself a circle, then is not unique and all solutions of are square-integrable. This is the limit-circle case.

Nonetheless, the differential operator

whose domain satisfies

where on the limit circle or limit point, is a self-adjoint differential operator (cf. also Self-adjoint operator; Self-adjoint differential equation) on .

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

The spectral measure of is given by

The spectral resolution of arbitrary functions in is

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

References

[a1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[a2]	A.M. Krall, " theory for singular Hamiltonian systems with one singular point" SIAM J. Math. Anal. , 20 (1989) pp. 644–700