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diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex new file mode 100644 index 0000000..1d772d6 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sturm.tex @@ -0,0 +1,56 @@ +% +% sturm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sturm-Liouville-Problem} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wellengleichung} +Saite mit variabler Massedichte führt auf die DGL +\[ +-y''(t) + q(t) y(t) = \lambda y(t), +\quad +q(t) > 0 +\] +mit Randbedingungen $y(0)=y(1)=0$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Sturm-Liouville-Operator} +\[ +A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p +\] +auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten +\[ +f(0)=f(1)=0 +\] +\end{block} +\end{column} +\end{columns} +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,Ag \rangle +&= +\langle f,-D^2 g\rangle + \langle f,qg\rangle += +- +\int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt ++\langle f,qg\rangle +\\ +&=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} ++\int_0^1 \overline{f}'(t)g'(t)\,dt ++\langle f,qg\rangle +=-\int_0^1 \overline{f}''(t)g(t)\,dt ++\langle qf,g\rangle +\\ +&=\langle Af,g\rangle +\end{align*} +\end{block} +\end{frame} +\egroup |