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Difference between revisions of "Titchmarsh-Weyl m-function"

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$$
 
$$
  
Now consider a real boundary condition at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012017.png" />, of the form
+
Now consider a real boundary condition at $b'$, $a<b'<b$, of the form
  
<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/t12012018.png" /></td> </tr></table>
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$$\cos\beta x(b')+\sin\beta px'(b')=0,$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012019.png" /> satisfy it. Then
+
and let $\chi(x,\lambda)=\phi(x,\lambda)+\mathrm l(\lambda)\psi(x,\lambda)$ satisfy it. Then
  
<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/t12012020.png" /></td> </tr></table>
+
$$\mathrm l(\lambda)=\frac{-(\cot\beta\phi(b',\lambda)+p\phi'(b',\lambda))}{\cot\beta\psi(b'(\lambda)+p\psi'(b',\lambda)}.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012022.png" /> is a [[Meromorphic function|meromorphic function]] in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012023.png" />-plane; indeed, it is a bilinear transformation. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012024.png" /> varies over real values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012026.png" /> varies over the real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012027.png" />-axis, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012028.png" /> describes a circle in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012029.png" />-plane.
+
If $z=\cos\beta$, $\mathrm l$ is a [[Meromorphic function|meromorphic function]] in the complex $z$-plane; indeed, it is a bilinear transformation. As $\beta$ varies over real values $0\leq\beta\leq\pi$, $z$ varies over the real $z$-axis, and $\mathrm l$ describes a circle in the $z$-plane.
  
It can be shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012030.png" /> increases, the circles become nested. Hence there is at least one point inside all. For such a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012031.png" />,
+
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 $\mathrm l=m(\lambda)$,
  
<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/t12012032.png" /></td> </tr></table>
+
$$\int\limits_a^b|\chi(x,\lambda)|^2w(x)dx<\infty.$$
  
There exists at least one solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012033.png" />, which is square-integrable.
+
There exists at least one solution of $Ly=\lambda y$, which is square-integrable.
  
If the limit of the circles is a point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012034.png" /> is unique and only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012035.png" /> is square-integrable. This is the limit-point case. If the limit of the circles is itself a circle, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012036.png" /> is not unique and all solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012037.png" /> are square-integrable. This is the limit-circle case.
+
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
 
Nonetheless, the differential operator
  
<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/t12012038.png" /></td> </tr></table>
+
$$Ly=\frac{-(py')'+qy}{w}$$
  
 
whose domain satisfies
 
whose domain satisfies
  
<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/t12012039.png" /></td> </tr></table>
+
$$\sin\alpha y(a)-\cos\alpha by'(a)=0,$$
  
<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/t12012040.png" /></td> </tr></table>
+
$$\lim_{x\to b}[p(x)(y(x)\chi'(\lambda,x)-y'(x)\chi(x,\lambda)]=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012041.png" /> on the limit circle or limit point, is a self-adjoint differential operator (cf. also [[Self-adjoint operator|Self-adjoint operator]]; [[Self-adjoint differential equation|Self-adjoint differential equation]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012042.png" />.
+
where $\mathrm l=m$ on the limit circle or limit point, is a self-adjoint differential operator (cf. also [[Self-adjoint operator|Self-adjoint operator]]; [[Self-adjoint differential equation|Self-adjoint differential equation]]) on $L^2(a,b;w)$.
  
If the circle limit is a point, the second boundary condition (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012043.png" />) is automatic.
+
If the circle limit is a point, the second boundary condition (at $b$) is automatic.
  
The [[Spectral measure|spectral measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012044.png" /> is given by
+
The [[Spectral measure|spectral measure]] of $L$ is given by
  
<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/t12012045.png" /></td> </tr></table>
+
$$\rho(\lambda)-\rho(\mu)=\frac1\pi\lim_{\epsilon\to0}\int\limits_\mu^\lambda\operatorname{Im}(m(\nu+i\epsilon))d\nu.$$
  
The [[Spectral resolution|spectral resolution]] of arbitrary functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012046.png" /> is
+
The [[Spectral resolution|spectral resolution]] of arbitrary functions in $L^2(a,b;w)$ is
  
 
<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/t12012047.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/t12012047.png" /></td> </tr></table>
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where the limit is in the mean-square sense, and
 
where the limit is in the mean-square sense, and
  
<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/t12012048.png" /></td> </tr></table>
+
$$g(\lambda)=\lim_{b'\to b}\int\limits_a^{b'}f(x)\psi(x,\lambda)dx.$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Krall,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120120/t12012049.png" /> theory for singular Hamiltonian systems with one singular point"  ''SIAM J. Math. Anal.'' , '''20'''  (1989)  pp. 644–700</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Krall,  "$M(\lambda)$ theory for singular Hamiltonian systems with one singular point"  ''SIAM J. Math. Anal.'' , '''20'''  (1989)  pp. 644–700</TD></TR></table>

Revision as of 16:35, 21 August 2014

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 $\chi(x,\lambda)=\phi(x,\lambda)+\mathrm l(\lambda)\psi(x,\lambda)$ satisfy it. Then

$$\mathrm l(\lambda)=\frac{-(\cot\beta\phi(b',\lambda)+p\phi'(b',\lambda))}{\cot\beta\psi(b'(\lambda)+p\psi'(b',\lambda)}.$$

If $z=\cos\beta$, $\mathrm l$ is a meromorphic function in the complex $z$-plane; indeed, it is a bilinear transformation. As $\beta$ varies over real values $0\leq\beta\leq\pi$, $z$ varies over the real $z$-axis, and $\mathrm l$ 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 $\mathrm l=m(\lambda)$,

$$\int\limits_a^b|\chi(x,\lambda)|^2w(x)dx<\infty.$$

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

$$Ly=\frac{-(py')'+qy}{w}$$

whose domain satisfies

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

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

where $\mathrm l=m$ 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

$$\rho(\lambda)-\rho(\mu)=\frac1\pi\lim_{\epsilon\to0}\int\limits_\mu^\lambda\operatorname{Im}(m(\nu+i\epsilon))d\nu.$$

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

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

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

References

[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: http://encyclopediaofmath.org/index.php?title=Titchmarsh-Weyl_m-function&oldid=33047
This article was adapted from an original article by Allan M. Krall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article