Chapter 6 Sturm–Liouville problems
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Definition 6.1. Let \(\a \in [-\infty ,\infty )\phantom {]}\), \(\b \in (-\infty ,\infty ]\) with \(\a <\b \) and let
\[ w,p,q:(\a ,\b )\to \mR , \]
by infinitely differentiable functions with \(w(x), p(x)>0\) for all \(x\in (\a ,\b )\). For \(\lambda \in \mC \), the Sturm–Liouville differential equation is
\[ (pu')'+qu=\lambda wu, \]
which can alternatively be written as
\[ pu''+p'u'+qu=\lambda wu. \]
We require that \(u\in L^2(\a ,\b ;w)\), where
\[ L^2(\a ,\b ;w):=\left \{u:(\a ,\b )\to \mR : \int _\a ^\b u(x)^2\,w(x)dx<\infty \right \}, \]
and \(L^2(\a ,\b ;w)\) is called the solution space and \(w\) is called the weight. In the special case that \(w=1\) we write \(L^2(\a ,\b )\) instead of \(L^2(\a ,\b ;1)\). The Sturm–Liouville differential expression is
\[ Du:=\frac {(pu')'+qu}{w}, \]
so that the Sturm–Liouville differential equation is \(Du=\lambda u\).
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Example 6.4. The differential equation for \(\Theta \) in polar coordinates obtained from the Laplacian (as in Chapter 5) which we recall equals
\[ u''=\lambda u, \]
with solution space
\[ L^2(0,2\pi ), \]
(we note that the weight equals 1 since the Jacobian determinant for polar coordinates doesn’t involve \(\theta \), see Section 4.2) corresponds to
\[ \a =0,\quad \b =2\pi ,\quad w=1,\quad p=1,\quad q=0. \]
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Example 6.5. The differential equation for \(R\) in polar coordinates obtained from the Laplacian (as in Chapter 5) we recall is (here \(\sigma \geq 0\) is a given parameter)
\[ u''+\frac {1}{x}u'-\frac {\sigma ^2}{x^2}u=\lambda u, \]
with solution space (this follows from Section 4.2)
\[ L^2(0,b;x). \]
We therefore see (from the solution space) that \(\a =0\), \(\b =b\) and \(w=x\). We then rewrite the differential equation as
\[ xu''+u'-\frac {\sigma ^2}{x}u=\lambda xu, \]
recognize that the right-hand side is now of the form \(\lambda w u\), so that the left-hand side should be \(pu''+p'u'+qu\). From the coefficient of \(u''\) we then see that \(p=x\) (which we check gives \(p'=1\) which is consistent with the coefficient of \(u'\)) and from the coefficient of \(u\) we see that \(q=-\frac {\sigma ^2}{x}\). Summarizing, this is a Sturm–Liouville problem with
\[ \a =0,\quad \b =b,\quad w=x,\quad p=x,\quad q=\frac {-\sigma ^2}{x}. \]
We note that the sign conditions \(w,p>0\) on \((\a ,\b )\) are indeed satisfied and that the smoothness condition on \(w,p,q\) on \((\a ,\b )\) is satisfied.
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Remark 6.6. To show that a second order linear differential equation with a given solution space \(L^2(\a ,\b ;w)\) is of Sturm–Liouville type, we have to do the following.
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• Read \(\a \), \(\b \) and \(w\) off from the solution space;
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• Write it in the form \(au''+bu'+cu=\lambda wu\);
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• Check that this is of the form \(pu''+p'u'+qu=\lambda wu\) (i.e. that \(a'=b\) in the notation of the previous bullet point);
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• Check that \(p,w>0\) on \((\a ,\b )\) (the sign condition);
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• Check that \(p,q,w\) are infinitely differentiable on \((\a ,\b )\) (the smoothness condition).
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