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Difference between revisions of "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|Self-adjoint differential equation]]). Begin with a formally symmetric differential expression
 
A function arising in an attempt to properly determine which singular boundary-value problems are self-adjoint (cf. also [[Self-adjoint differential equation|Self-adjoint differential equation]]). Begin with a formally symmetric differential expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201202.png" /></td> </tr></table>
+
$$
 +
L y = \frac{-(p y')' + q y}{w} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201204.png" /> are measurable coefficients over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201205.png" />, and which is defined on a domain within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201206.png" />. The Titchmarsh–Weyl <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201207.png" />-function is defined as follows: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t1201209.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012011.png" /> be solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012012.png" /> satisfying
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012013.png" /></td> </tr></table>
+
$$
 +
\begin{aligned}
 +
  \phi(a,\lambda) &= \sin\alpha, & \psi(a,\lambda) &= \cos\alpha, \\
 +
  p\phi'(a,\lambda) &= -\cos\alpha, & p\psi'(a,\lambda) &= \sin\alpha .
 +
\end{aligned}
 +
$$
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012014.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012014.png" /></td> </tr></table>

Revision as of 12:44, 29 October 2012

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
How to Cite This Entry:
Titchmarsh-Weyl m-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Titchmarsh-Weyl_m-function&oldid=28680
This article was adapted from an original article by Allan M. Krall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article