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
$$ \begin{aligned} \phi(a,\lambda) &= \sin\alpha, & \psi(a,\lambda) &= \cos\alpha, \\ p\phi'(a,\lambda) &= -\cos\alpha, & p\psi'(a,\lambda) &= \sin\alpha . \end{aligned} $$
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Now consider a real boundary condition at ,
, of the form
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and let satisfy it. Then
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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
,
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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
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whose domain satisfies
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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
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The spectral resolution of arbitrary functions in is
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where the limit is in the mean-square sense, and
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References
[a1] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) |
[a2] | A.M. Krall, "![]() |
Titchmarsh-Weyl m-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Titchmarsh-Weyl_m-function&oldid=14178